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Introduction to Modern Physics 

By 

Charles W Fay 



Introduction to Modern Physics: Physics 311 

Lecture Notes 
Ferris State University 
unpublished supplement to the text 



Department of Physical Science 
2011 



Table of Contents 



I The Birth of Modern Physics 1 

1 What is Physics 2 

1.1 Classical Physics and Modern Physics 2 

1.2 Assumptions of Science 3 

2 Classical Mechanics 5 

2.1 Physics Before the 19th Century 5 

2.2 Experiment and Observables 6 

2.3 Inertial Reference Frames (IRF) 9 

2.3.1 Symmetries 9 

3 Classical Electro-Magnetism 10 

3.1 Maxwell and Electro-Magnetism 10 

3.1.1 Maxwell's Equations in the integral form 10 

3.1.2 Maxwell's Equations in the derivative form 10 

3.2 Reference Frames? 12 

3.3 Implications of the Speed of Light 12 

3.4 Vector Calculus 12 

3.5 EM and Optics 13 

II Relativity 14 

4 Special Relativity I 15 

4.1 Two Postulates 15 

4.1.1 Simultaneity 16 

4.1.2 Measurement of Length 16 

4.1.3 Simultaneity in moving frames 17 

4.2 Building a suitable transformation: Taken from Relativity by Einstein . . 17 

4.3 Length Contraction 20 

4.3.1 Example: Length Contraction 21 

4.4 Time Dilation 21 

4.4.1 Example: Time Dilation 21 

4.4.2 Graphical derivation of Time Dilation 22 

4.4.3 Proper length and proper time 22 

4.4.4 Time discrepancy of an event synchronized in K as viewed in K' sepa- 

rated by X2 — x\ 23 

4.4.5 Time discrepancy of events separated by At and Ax in K as viewed by 

K' 23 

ii 



4.5 Doppler Effect 23 

4.5. f Example: Doppler Effect 25 

4.6 Twin Paradox 25 

5 Special Relativity II 26 

5.1 Velocity Transformations 26 

5.1.1 Example: transforming the velocity of an object moving at the speed of 

light 27 

5.1.2 Transformation of the angle of a light ray 28 

5.2 Transformation of the Energy of Light Rays 28 

5.3 Does the Inertia of a Body depend upon its Energy-content? 29 

5.4 Momentum 30 

6 Four vectors and Lorentz invariance 33 

6.1 Momentum 33 

6.1.1 Invariant interval 34 

6.1.2 four- velocity 37 

6.1.3 Four-momentum 37 

6.1.4 Conservation of Momentum 38 

6.1.5 Classical momentum conservation under a Galilean transformation . . 38 

6.1.6 Relativistic Momentum Conservation 39 

6.1.7 Particles of zero mass 41 

6.2 Mechanical Laws 42 

6.2.1 Motion under a constant force 43 

6.2.2 Cyclotron Motion 43 

6.3 Proper Force 44 

7 General Relativity 45 

7.1 Absolute Acceleration? 45 

7.2 2 principles of General Relativity 45 

7.2.1 Principle of Covarience 46 

7.2.2 Results of a transforming rotation in S.R 46 

7.2.3 Principle of Equivalence 47 

7.3 Experimental support for General Relativity 48 

7.3.1 Black Holes 48 

III Origins of Quantum Mechanics 50 

8 QMI: Quantum Hypothesis 51 

8.1 The energy density 53 

8.1.1 number density of modes in a cavity 54 

8.1.2 Solution by Micro-canonical Ensemble 55 

iii 



8.1.3 Solution by canonical ensemble 57 

8.2 Einstein and the Photoelectric Effect 59 

8.2.1 The Photoelectric Effect 60 

9 QMII: The Atom 65 

9.1 The Electron 65 

9.1.1 J.J. Thomson discovers the electron 65 

9.1.2 Fundamental Charge: Robert Millikin 69 

9.1.3 Measuring M by terminal speed 70 

9.2 The Nucleus: Rutherford 71 

9.2.1 The Rutherford Formula 72 

9.3 The Bohr Model 75 

9.3.1 The Bohr radius 76 

9.3.2 The energy of the Bohr atom 77 

9.3.3 Generalizing the Bohr Model 79 

9.4 The Neutron 80 

10 QM III: Elements of Quantum Theory 81 

10.1 Compton Scattering 81 

10.1.1 Example: Compton Scattering 83 

10.2 de Broglie 84 

10.2.1 Example: de Broglie wavelength 84 

10.2.2 Quantization of angular momentum 85 

10.3 Wave-particle duality 85 

10.3.1 Young's Double slit Experiment 85 

10.4 Probability Waves 86 

10.4.1 Example: Probability density 87 

10.5 Electron wave packets 87 

10.5.1 Heisenberg Uncertainty 89 

10.5.2 Schroedinger's Equation 90 

10.5.3 Particle in a one-dimensional box 91 

10.5.4 Tunneling 93 

10.6 Spin 93 

10.7 The Copenhagen Interpretation 94 

11 QM IV: Atomic Structure 95 

11.1 Multi-electron Atoms 96 

11.1.1 Pauli Exclusion Principle 96 

11.2 Periodic Table 96 

11.3 Heisenberg Uncertainty 97 

11.4 Relativistic Quantum Mechanics 98 

11.5 Nuclear Structure 98 

11.5.1 Nuclear Force 99 

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11.6 Nuclear Notation 99 

11.7 The Laser 100 

11.7.1 Absorption and Spontaneous Emission 100 

11.7.2 Holography 100 

12 QM V: Radiation 101 

12.1 Radioactivity 101 

12.1.1 Alpha Decay 101 

12.1.2 Example 102 

12.1.3 Beta Decay 102 

12.1.4 Gamma Decay 104 

12.1.5 Example 105 

12.1.6 Radiation Penetration 105 

12.2 Half-life 106 

12.2.1 Example: Sr-90 106 

12.2.2 Radioactive Dating 107 

12.2.3 Example: Carbon 14 dating 108 

12.3 Nuclear Stability 108 

12.3.1 Binding Energy 110 

12.4 Radiation Detection Ill 

12.4.1 Gieger Counter Ill 

12.4.2 Scinillation Counter Ill 

12.4.3 Semi-conductor Detector Ill 

12.4.4 Cloud Chamber 113 

12.5 Biological Effects and Medical Applications 113 

12.6 Nuclear Reactions 115 

12.6.1 Conservation of mass-energy 115 

12.6.2 Example 116 

12.7 Nuclear Fission 116 

12.7.1 Nuclear Power 117 

12.7.2 Breeder Reactor 119 

12.7.3 Safety 119 

12.8 Nuclear Fusion 119 

12.8.1 Example: Fusion and the sun 120 

12.8.2 Fusion Power 120 

12.9 Neutrino 120 

13 QM VI: The Standard Model 122 

13.1 Beta Decay and neutrino 122 

13.2 Fundamental Forces 122 

13.3 Electro-Magnetic force and the Photon 123 

13.4 Strong Force and mesons 123 



v 



13.5 Weak nuclear force - W particle 124 

13.6 Gravity 124 

13.7 Fundamental particles 125 

13.8 Unification Theories 126 

IV Applications of Quantum Mechanics 127 

14 Quantum Statistics 128 

15 Semiconductors 129 
APPENDICES 130 
A Nobel Prizes in Physics 131 

B Maxwell's Equations in the integral form 136 

B. 0.1 Maxwell's Equations in the derivative form 136 

C Alternate Derivations for Special Relativity 138 

C. l Building a suitable transformation: Einstein's way 138 

C.l.l The nature of <f>(v) 142 

C.2 The Lorentz transformation as a rotation in the complex plane 143 

C.3 Velocity Transform 145 

C.3.1 Example: transforming the velocity of an object moving at the speed of 

light 147 

C.3. 2 Transformation of momentum and energy from one frame to another . 147 



vi 



List of Papers 
Introduction to Modern Physics 

1. On the electrodynamics of moving bodies (1905) Albert Einstein 

2. Does the inertia... (1905) Albert Einstein 

3. On the generalized theory of gravitation (1950) Albert Einstein 

4. Cathode Rays [the discovery of the electron] (1897) J. J. Thompson 

5. On the Law of Distribution of Energies in the normal spectrum (1901) Max 
Plank 

6. On a Heuristic Point of View about the Creation and Conversion of Light (1905) 
[photoelectric effect] Albert Einstein 

7. The Elementary Electric Charge (1911) Robert A. Millikan 

8. The Scattering of alpha and beta particles by Matter and the Structure of the 
Atom [Nucleus] (1911) E. Rutherford 

9. On the Constitution of Atoms and Molecules (1913) Neils Bohr 

10. The Structure of the Atom (1914) E. Rutherford 

11. A Quantum Theory of the Scattering of X-rays by Light Elements (1923) Arthur 
H. Compton 

12. Radiation (1923) de Broglie 

13. Heisenberg (1925) 

14. Schrodinger (1926) 

15. Radiations from Radioactive Substances (Neutron)- James Chadwick (1932) 



Part I 

The Birth of Modern Physics 



i 



Chapter 1 
What is Physics 



Physics is the study of matter, energy and change. The physical world is studied 
through experimentation. It is through experimentation that all physical theories 
are tested and confirmed. Unlike math, its close cousin, Physics only makes sense in 
relation to what occurs in the possible in the physical world. A physical theory may 
be elegant or satisfying but it if does not conform to events then it is of no value. 

One example can be found in electrostatics. Electrostatic fields are dependent 
upon the separation of positive and negative charges. Each charge exists as either a 
positive or negative monopole. If two charges are placed on either end of an imaginary 
rod, we can create what is known as a dipole. Magnetostatics would be more satisfying 
if there were individual magnetic " charges" . Magnets only come in dipoles each south 
pole is accompanied by a north pole. If we could isolate either magnetic pole. Then 
magnetostatics would be formulated exactly the same as electrostatics. However, 
magnetic monopoles are not found in nature. 

Mathematics provides a frame work and a language for physics. In applying 
math it needs to be remembered that physics is an experimental science. Often 
times math gives an avenue for theorists. As an example we are familiar with 3 
physical dimensions (x,y,z) but we might imagine 4, or 5, or 10 or a million similar 
dimensions. One consequence of Einstein's special relativity is the addition of time 
as another dimension. Instead of space existing in 3 dimension (x,y,z), we have a four 
dimensional space-time system, (x,y,z,t). 

1.1 Classical Physics and Modern Physics 

Classical Physics is the name we give to the physical theories generated before the 
twentieth century. Classical physics is comprised primarily of two areas, mechanics 
and electro-magnetism. Understanding how physics progressed to the point where 
modern physics begins is important to understanding modern physics. Many of the 
concepts in modern physics are extensions of ideas that find their origin in classical 
physics. 



2 



Chapter 1 Introduction to Modern Physics: Physics 311 

Modern Physics is the name we give to the Theoretical advancements in physics 
since the beginning of the 20th Century, popularly known as Relativity and Quan- 
tum Mechanics. In this book, we will examine some of the seminal theoretical and 
experimental papers that formed the basis of modern physics. 

Some concepts that will be developed include, 

• The Special and General theories of Relativity 

• The electron 

• Quantization of energy 

• Photo-electric effect 

• The Bohr Atom and spectroscopy 

• Compton Scattering 

• Electron Free and Bound states 

• Atoms in a Magnetic Field 

• Heisenberg 

• Schrdinger's Equation 

• The Copenhagen interpretation of Quantum Mechanics 

• Quantum Statistics 

• Semiconductors 

• The Standard Model 

1.2 Assumptions of Science 

In order to have a meaningful science we need to start with three assumptions. These 
assumptions are important to experimentation. 

1. Common Perception 

2. Uniformity of space and time 

3. Natural Causes 



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The first assumption common perception, means that two independent observers 
observing the same experiment under the same conditions with agree on their obser- 
vations. To facilitate agreements these observations must be quantified. 

Uniformity of space means that physical laws are the same regardless of the po- 
sition. It is possible that two experiments under different conditions may appear 
different, but we assume that the rules are the same and it is only the conditions of 
the experiment and the the observation that change the outcome. 

The last assumption is strictly speaking a limitation on science that first two 
assumptions describe assumptions about the nature of the systems studied. The 
assumption of natural causes limits science to describing those phenomena that are 
the result of the nature of the universe. Supernatural events, those events that are 
outside of the universe cannot be studied since there are singular and therefor not 
repeatable. 

Some scientists take this assumption to the point where there is assumed to be 
nothing outside of the universe. Since this cannot be shown to be true or false by 
experimentation. It is a theological and not a scientific statement. 



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Chapter 2 
Classical Mechanics 



2.1 Physics Before the 19th Century 



One might say the golden era of science really began with the Galileo, who was 
ushered in the area of the experimental scientist. Before Galileo, science was largely 
philosophical, meaning theory was neither the result nor accompanied experiment. 
The result of this was a theories that were speculative and limited in their ability 
to explain physical phenomena, (for an excellent summary of the development of 
classical physics see "Introduction to Modern Physics by Richtmyer, et. al. 1969). 

After Galileo, the giant who dominated physics for nearly 200 years (and is very 
important today) was Isaac Newton. Newton, among other things, was the first to 
unify the theory of motion and gravitation. Newton linked the acceleration a body 
feels on the earth (apocryphally an apple) to the motion of celestial bodies. 

Newton proposed a simple set of principles; 

1. Law of Inertia: (also known as Galileo's Law of Inertia) An object 
in constant motion stays in constant motion unless acted on by an outside 
unbalanced force. 

2. The acceleration of an object is proportional to the net force applied to the 



The proportionality constant is a quantity we will define as the mass, m. 
3. For every force there is an equal and opposite force, 



object. 



F = ma 



(2.1) 



F12 — —F 2 \ 



(2.2) 



4 



Newton's law of gravitation can be written as, 



F — —G 



mim 2 



(2.3) 



5 



Chapter 2 



Introduction to Modern Physics: Physics 311 



Which simply states that two bodies of masses, m 1 and m 2 j exert a force on each 
other proportional to the product of their masses and the inversely proportional 
to the square of the distance between their mass centers. 

Newton also did work in optics and mathematics. 



2.2 Experiment and Observables 

Our understanding of the physical world is predicated on those quantities which can 
be observed under the controlled environment of experimentation. We refer to these 
as quantities as observables. 

All measurements are made relative to a system of coordinates. When an exper- 
iment is performed we make arbitrary choices about where the origin is placed and 
when a clock is started. This is known as the reference frame. A frame of reference 
(or reference frame) is a set of coordinates (x,y,z) that define the position of a body. 

In the case of a train, we might define a reference frame that is stationary with 
respect to the train. We would then observer any object sitting on the train is 
stationary. However, we might define a reference frame that is stationary with respect 
to the platform. Now, the objects that are stationary with respect to the train are 
moving with respect to the platform. Similarly this effects bodies that are moving 
with respect to the train If a ball is dropped on the train to an observer on the train 
the ball appears to fall straight down, while an observer on the platform sees the ball 
travel in a parabolic path. 

For the observer on the platform the motion is the vector sum of the motion of ball 
relative to the train and the train relative to the platform. Classically, the principle 
of relative motion was defined by Galileo, 

• Galilean Relativity: When two observers are in constant relative motion 
(^relative = constant) to each other they will observe the same physical laws. 

The uniformity of space implies homogeneity, and that physical laws should remain 
the same under translation and rotation. In physics an objects position at a given 
instant is specified by a point of three coordinates pi = (x, y, z) The laws of mechanics 
then summarize as, 

d 2 x „ d 2 y „ d 2 z „ 

If a series of phenomena are measured from two points the two observers should 
agree on the same laws. Each observer will measure the experiment from their own 
coordinate system. 

We want to show that a transformation from one system to another will not change 
the laws of mechanics. For simplicity we will talk about two systems that have their 



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Chapter 2 Introduction to Modern Physics: Physics 311 

axes aligned parallel, K (x,y,z) and K'(x', y', z'). To transform from K to K', 

x' = x — a,y' = y, z' = z (2.5) 

In order for Newton's laws to remain unchanged undertranslation, 

F x > = F x , F y i = F y , F z i = F z (2.6) 

By inspection we see that the last two remain the same. Looking at the transforma- 
tion, 

x' = x - a (2.7) 

Applying a variation in time, 



dx' d(x — a) dx da 

dt dt dt dt 

since a is a constant da/dt = and, 

dx' dx 

~dt ~ ~dt 

and correspondingly, 

d 2 x' d 2 x 



(2.8) 



(2.9) 



(2.10) 



dt 2 dt 2 

F X ' = Fx (2.11) 

We say that the laws of physics are symmetric under translation. 

Lets examine rotation. The points in K' rotated in the x,y plane at an angle of 9 
relative to K. 

x' = x cos 9 + y sin 9 (2.12) 

y' = ycos9 — xsin9 (2.13) 

z = z. (2.14) 

simply the force F in K' is given by, 

F X ' = F x cos9 + F y sin9 (2.15) 

F y , = Fy cos 9 - F x sin 9 (2.16) 

F Z ' = F z . (2.17) 

Now calculate d 2 x/dt 2 when 9 is a constant, then, 

d 2 x' d 2 x „ d 2 y . n , n 

§ - ,, 19 ) 



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Chapter 2 



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Multplying by the mass and we clearly get, 

F x , = F x cos + F y sin (2.20) 
Fy> = F y cos 6 - F x sin 9 (2.21) 

Newton's laws are symmetric under rotation. 

Under a Galilean transformation a body moving with a speed v in reference frame 
S, observed from reference frame S' moving with speed v' in relation to S, will appear 
to move at the speed, 

u = v - v' (2.22) 

If S' is moving with speed — v' (moving in opposite direction of S). The speed of the 
body will appear to be, 

u = v + v' (2.23) 

Mathematically, this is a simple transformation from one coordinate system S to 
another S', a point in S' can be related to a point in S by the boost, 

x' = x - v't (2.24) 

Now, 

doc dx , _ „ _ , 

M = H- V (2 ' M) 
d 2 x' d 2 x 

Newton's laws are symmetric under a galilean boost. 



Homework 1 

Show how Newton's law of inertia is violated for a reference frame where two bodies 
are in free-fall but at rest with each other. 

From the homework problem it can be shown that Newton's Laws are consistent 
for boosts corresponding to constant velocities. In Fact, the definition of just such a 
reference frame borders on the philosophical. What is meant by a constant velocity, 
since we know that each measurement must be made relative to some reference frame. 
Newton defined an absolute rest frame in which his laws hold true, then for any other 
reference frame in constant relative motion to that absolute frame, the physical laws 
would true, (absolute motion is motion with respect to absolute space, a purely 
theoretical concept, while relative motion is motion relative to some chosen object) 
Similarly, Newton defined absolute time. 

Now, all experiments are done in some relative reference frame since we cannot 
determine and/or do not know the "absolute rest frame". Since the law of inertia can 
be shown to be true by experiment, there must exist reference frames for which the 
laws of mechanics hold, in particular the law of inertia. These reference frames are 
known as inertial reference frames. 



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Chapter 2 



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2.3 Inertial Reference Frames (IRF) 

1. The law of inertia can be verified by experiment in an IRF. 

2. Frame is not accelerating relative to other IRFs. 

3. Any frame which is moving with uniform velocity relative to an IRF is an IRF 

This leads the a set of transformations between a point (x, y, z) in one inertial 
frame S and a point (x',y',z') in another inertial frame S'. Where S' is moving with 
speed v' relative to S. (align the reference frames such that all axes are parallel to the 
respective primed axis and v' is on the x,x' direction.) 

x < = x-v't (2.27) 
V = y' (2.28) 
z' = z (2.29) 
t' = t (2.30) 

(2.31) 

To be complete we include the time in the transformation as we need to know the time 
a particle is at a particular position in order to fully understand its motion. This set 
of transformations is known as a Galilean transformation or a boost transformation. 

The full set of any transformations must include translations (as listed above) 
and rotations. Newton's laws must be invariant under both. It can be shown that no 
mechanical experiment can detect any intrinsic difference between inertial reference 
frames. 



2.3.1 Symmetries 

So far we examined transformations related to translation, boosts and rotations. All 
3 showed what we called symmetries. Symmetry is important to physics, in quantum 
mechanics a symmetry corresponds to a conservation law. 

symmetry conservation law 

translation in space momentum 
translation in time Energy 
Rotation through a fixed angle angular momentum 

Uniform velocity in a straight line (Lorentz transformation) 



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Chapter 3 

Classical Electro-Magnetism 



3.1 Maxwell and Electro- Magnetism 

The next major unifying work was done by James Clerk Maxwell, who brought to- 
gether different experimental and theoretical of electricity and magnetism. I don't 
want to slight the great work by physicists between Newton and Maxwell. Maxwell 
is notable however for combining the work of Ampere, Gauss and Faraday, into a set 
of equations that now bare Maxwell's name. Maxwell's equation's also showed that 
the electric and magnetic fields traveled in waves at the speed of light. 

3.1.1 Maxwell's Equations in the integral form 




(3.1) 



(3.4) 



(3.2) 



(3.3) 



3.1.2 Maxwell's 



Equations in the derivative form 



1 



(3.5) 
(3.6) 
(3.7) 



V-E = 



P 



V -B 



e 




Vx£ = = - 



dB 
~dt 



V x B = /j J + n e 



BE 
~dt 



(3.8) 



10 



Chapter 3 Introduction to Modern Physics: Physics 311 

in regions where there is no charge or current, we write Maxwell's equations as, 

V-E = 
V-B = 
„ „ dB 

Vx£ = =~m 

„ „ dE 

V x B = Hoeo-T^r 

Now we apply the operator Vx to the second two equations to get, 

pjTD 

V x (V x E) = V(V • E) - V 2 E = Vx(— ) 

d ,„ d 2 E 

d 2 E 



since V • E — 0; 

and similarly, 

Vx(Vx5) = V(V-5)-V 2 5 = Vx(-) 

since V • B — 0; 

<9 2 .B 

V 2 B = Voto-r^ 

Which is the same as the wave equation, 

v/ = 1^ 

the solution of which is, 

/ = / sin(£;:r - tot) 

Where the velocity v, 

1 

is same as the speed of light c = 3.00 x 10 8 m/s, and 

to = 2irf 

The discovery that maxwell's constant c for electromagnetic waves was the same as 
the speed of light in a vaccum, lead to the conclusion that light was an electromagnetic 
wave of the same form as x-rays, and radiowaves, just on different wavelenths. 



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Chapter 3 



Introduction to Modern Physics: Physics 311 



3.2 Reference Frames? 

One difficulty between Newtonian Mechanics and Maxwell's equations are the com- 
plex relationship between Electric and magnetic fields. Imagine a magnet in motion 
and a conductor at rest, this results in a electric field with an associated energy and 
a current in the conductor. While a simple change of reference gives a stationary 
magnet and moving conductor, which leads to no Electric field (and an abscence of 
associated energy) + an EMF in the conductor and a current. 

We note that Maxwell's equations are independent of direction and propogation 
and contain no reference to the frame in which the motion happens, In fact exper- 
imentation shows that reguardless of the frame in which Maxwell's equations are 
measured. The results are the same, and the speed of light is measured at the same 
value. How can this be as the theory of Inertial reference frames (Newtonian Me- 
chanics) implies that the value of the speed of light should vary by the speed of the 
relative frames, it doesn't. 

• What does the fact that Maxwell's equations holds for all inertial frames imply? 

• What does this mean to the concepts of space and time? 

The implication is that the speed of light is the same for all inertial frames. We have 
to ask now is there a transformation for which Newton's laws hold, and the speed of 
light is constant for all inertial frames? 



3.3 Implications of the Speed of Light 

1. As yet unknown medium for light to travel through 

2. There is no medium and light travels with a constant speed requarless of refer- 
ence frame. Requires a new relativity. 



3.4 Vector Calculus 

Gradiant, 



Divergence, 



*f = sr s + ¥> + t k < 3 - 24 > 

ox ay oz 

v ^f + f + i < 3 - 25 > 

Curl 

V x^<f-f)Mf-f>^f -%)t (3.26) 



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Chapter 3 Introduction to Modern Physics: Physics 311 

Laplacian. 



Homework 2 

Show that Maxwell's equations do not hold under a boost transformation. 



3.5 EM and Optics 



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Part II 
Relativity 



14 



Chapter 4 
Special Relativity 



Read: On the Electrodynamics of Moving Bodies, Does the Inertia of a 
Body depend upon its Energy-content? 

It has been shown, that the equations governing electrodynamics do not transform 
via Galilean transformations. We ask the question is there a transformation for 
which the laws of mechanics and electrodynamics hold true? This question was first 
answered by Lorentz, who constructed a mathematical transformation, that satisfied 
the requirement of maintaining the structure of electrodynamics. There was however 
no physical support for such a transformation. 

In 1905, Albert Einstein provided the physical support for the Lortenz transfor- 
mation and in his seminal paper, "On the Electrodynamics of Moving Bodies" (1905), 
which introduced the concept of special relativity. 

In the discussion of Einstein's paper we will use for clarity his notion with differs 
from the standard notation in use for a transformation between one frame K given 
by the coordinates (x, y, z, t) and another frame K' given by (£, rj, (, r). 

4.1 Two Postulates 

Einstein postulated that physical theory should be mediated by two conditions. 

1. (Principle of Relativity) Laws by which the states of physical systems undergo 
change are not affected, whether these changes of state be referred to the one 
or the other of the two systems of coordinates in uniform transitory motion. 

2. Any ray of light moves in the "stationary" system of coordinates with the de- 
termined velocity c, whether the ray be emitted by a stationary source or a 
moving one. 

The first postulate states that any physical law should be stated in such a way that 
definition of a reference frame is not necessary for its validity. (In actuality the first 



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Chapter 4 



Introduction to Modern Physics: Physics 311 



postulate is limited to coordinate systems that are moving with constant relative 
motion. Einstein left it to general relativity to include accelerating frames.) 

The second postulate states a physical law as the constraint, namely that the 
anything moving with the speed c in one frame will be measured as moving at the 
speed c in all reference frames. This constraint is necessary to ensure that Maxwell's 
equations are acceptable in the this new formalism. 

Since physics is an experimental science, theories must be experimentally verified, 
so we take a moment to discuss measurements. To measure the length of an object I 
hold a rule to it and measure the positions of the edges of the object relative to the 
rule. The measurement is made by judging the position of the rule and the position 
of the object simultaneously. 

4.1.1 Simultaneity 

All judgments of time are judgments of simultaneous events. "A train arriving at 
7:00 really means the pointing of the small hand of my watch to 7 and the arrival of 
the train are simultaneous." 

It is easy to see that as events become more distant it is difficult to evaluate times. 
Because the speed of light is finite, we cannot synchronize with a distant event. Thus 
at point A we can measure events near A and at point B we can measure events near 



In a frame, where a clocks A and B are stationary with respect to each other the 
clocks are synchronized if the time required for light to travel from A to B (tg — t^) 
is equal to the time required to travel from B to A. (t' A — ts), 



1. if A synchronizes with B, B synchronizes with A. 

2. If A synchronizes with B and C, B synchronizes with C. 
Thus the speed of light is given by, 



Why do we spend time on the concept of measurement and simultenaity? The 
reason is simple .... 

4.1.2 Measurement of Length 

How does our view simultaneity effect the measurement of length. 

1. An observer moving with a measuring rod and the rod which is to be measured, 
conclusion: The rod measures /, the same as if it is at rest. (This is simply an 
expression of the principle of relativity.) 



B. 



tB — tA — t A — t B . 




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2. A rod moving relative to an observer and a measuring rod. Conclusion: the 
length of the rod is not measured as I. Because of the finite speed of light the 
ends of the rod are not measured simultaneously, therefore the rod will appear 
smaller. 



4.1.3 Simultaneity in moving frames 

[INSERT GRAPHICAL REPRESENTATION] 

Only observers at rest will agree w/ one another on the simultaneity of two events. 
The condition of simultaneity becomes, 

l -(t' A + t A )=t B (4.3) 

Imagine a light pulse leaving at t A = 5 and returning at t' A = 10, in the frame where 
the clocks are at rest with each other the light pulse reflects at a time halfway between 
t' A and tA or 7.5, From the condition of simultaneity we find, 

ta = \(t' A + t A ) (4.4) 

t B = ^(10 + 5) = 7.5 (4.5) 

(4.6) 

Here WE NEED TO INCLUDE A DISCUSSION OF TIME 



4.2 Building a suitable transformation: Taken from 
Relativity by Einstein 

Again we take 2 reference frames K and K' that are in K' is moving with velocity v 
relative to K. 

If at time t = r = 0, a pulse of light is sent out along the x— axis, from x = £ = 
y = i] = x = ( = 0, then the position of light pulse may be described by, 



which can be written as, 
Similarly in the K', 



x = ct (4.7) 
x - ct = (4.8) 



C = cr (4.9) 
i-cr = (4.10) 



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Chapter 4 Introduction to Modern Physics: Physics 311 

Since this one event must simultaneously satisfy both equations (4.8) and (4.10), 
they must satisfy, 

£ — cr = X(x — ct) 
similarly for a pulse moving in the —x direction, we obtain the relation, 

£ + cr = X(x + ct). 

Using equations (4.11) and (4.12) solve for £ and r, 

£ = ^x-^ct, 
s 2 2 

A + /i j A - /i 

cr = cr x. 

2 2 

since A and /j, are arbitrary functions, we will substitute, 

A + /! 



7 

r 



2 ' 
A — ji 



Allowing us to write £ and cr as, 

£ = ix -Vet, 
ct = jet — Tx. 

Looking at the origin of K' where £ = gives, 

Tc 

x = — t 

7 

Since this point must have a position of, 

x = vt. 

This then gives the velocity of K' relative to K, 

Tc 



v = — 

7 



Now rewrite equations (4.17, 4.18), 



£ = jx-'yvt, 
v 

ct = jx — 7-i. 

c 



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Chapter 4 Introduction to Modern Physics: Physics 311 

In order to determine the function a, imagine a frame if' that is at moving with 
velocity v relative to frame if, the transforms of x and t are now given by, 

f = i{x-vt), (4.24) 
r = >y(t-^x). (4.25) 

Further, there is a frame K"(x', y', z', t') that is moving relative to if' with a velocity 
of — v, then the transform of £ and r into if" is given by, 

x' = -y^ + vr), (4.26) 

if = l(r+~ 2 0- (4-27) 
cr 

Then transform from if to if" is given by, 

x' = 7(7(0; - ui) +7«(t- ^x), (4.28) 



*' = 7 ( 7 (t- -x) + 7 -(x-t;t)). (4.29) 



Simplifying, 



c 2 c 2 



w 2 

a:' = 7 2 (x-wt + wt- — x), (4.30) 

c 2 



t' = l 2 (t--x+-x--t), (4.31) 



and, 



x' = 7 Ml-^), (4-32) 

cr 



f = 7 2 t(l-^), (4-33) 

Since if and if" must be at rest with respect to each other, x — x f , and t = t', a 
must satisfy, 

1 = 7 2 (1-^), (4.34) 

cr 

7 = -4=. (4-35) 
'l-|r 



Allowing us to rewrite our transforms as, 



X — Vt / , „„N 

£ = ; (4-36) 



c 

t- ^ 



1 - 4 



(4.37) 



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for simplicity we introduce /3 — - 



7 = , (4-38) 

t = l(x-Pct), (4.39) 

r = 1 {t--x). (4.40) 

c 

Since the motion is orthogonal to y and z they cannot be affected by the relative 
velocity so the transforms above are supplemented by, 

V = V (4-41) 

C = z (4.42) 

We can see that a necessary condition of the second postulate of relativity (con- 
stancy of the speed of light) is that a light pulse in 3 dimensions forms a spherical 
wave front, if it propagates from the origin we write, 

r = ^Jx 2 + y 2 + z 2 = ct (4.43) 

x 2 + y 2 + z 2 - cH 2 = (4.44) 

The same light wave in K' \ 

r> = v^ 2 + V 2 + C 2 = cr (4.45) 

£ 2 + r] 2 + C 2 - c 2 t 2 = (4.46) 

They must satisfy the relation, 

e + v 2 + C 2 - c 2 r 2 = a{x 2 + y 2 + z 2 - cH 2 ) (4.47) 



Homework 3 

Show the derived transformation holds for a = 1. 



4.3 Length Contraction 

Let us consider how the transformation of two points leads to a change in length, and 
a change in time. Consider a rod of length L p = £ 2 — £i at rest in frame K' . 



6 = 7(zi-vt), (4.48) 

6 = i{x 2 -vt), (4.49) 

6-6 = -y(x 2 - Xl -(vt-vt)), (4.50) 

6-6 = 7(^2 -xi), (4.51) 

L p = 1 L. (4.52) 



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Chapter 4 Introduction to Modern Physics: Physics 311 

Where L p is the length of the rod as measured in its rest frame, and L is the length 
as measured in frame where the rod is moving. The length of a moving rod is shorter 
than its rest length. 

4.3.1 Example: Length Contraction 

If a rod is lm at rest and moving at a speed 0.8c, what is the new length of the rod? 



given: L p = lm v = 0.8c v/c = 0.8 7 = 1/^/1 - £ 

L p = (4.53) 



L n L v 2 



7 V c 
= (lm)y/l - (0.8) 2 = 0.6m (4.55) 

4.4 Time Dilation 

Now examining the difference between 2 times, consider two times T\ and r 2 happening 
at the same place £' in K', 

h = 7(ti + 40, (4-56) 
c z 

*2 = 7(r 2 + 40, (4-57) 
c z 

At = t 2 -t 1 = 7(T-2-r 1 )+7(^ , -^ , ), (4-58) 
At = t 2 -t 1 = 7AT = 7At p . (4.59) 

Because the events happening at Ti and r 2 occur at the same place, they are stationary, 
this time difference is referred to as the proper time, t p . The time At as measured 
from a frame where the there is movement between the two events the time measured 
is slower than the proper time. 

4.4.1 Example: Time Dilation 

If a stationary event takes 1.00s in K' and K' is traveling at v — 0.8c in relation to 

frame K, find the time the event takes in frame K. 

given: At p = Is v = 0.8c = /c = 0.8 (5 = 1/y/l - (5 2 

At = -/At„ = - 1 t v (4.60) 

' 1.0s) = 1.7s (4.61) 



1 - (0.8) 



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4.4.2 Graphical derivation of Time Dilation 

The time it takes like to strike a mirror as distance d away and return is, 

At = — (4.62) 

c 

If the source and mirror are now placed in a ship moving perpendicular to the 
direction of the light at a speed of v, The time the light takes will appear as, 

2D 

At' = — (4.63) 

c 

where, 



vAt' 

D = \{d? + {—f (4.64) 



combine this with Eq. (4.63) gives, 



(^-) 2 = d 2 + ( — ) 2 (4.65) 

,cAt. 9 r vAt' ,n 
= (^-) 2 + (^-) 2 (4-66) 



2 



,2 „2 



(-)At' 2 (l--) = ( C -)At* (4.68) 

At /2 (1-^) = At 2 (4.69) 
At' 2 = 7 At 2 (4.70) 

4.4.3 Proper length and proper time 

The proper length of a rod is the length of a rod measured in its rest frame with a 
ruler that is at rest with respect to the rod. 

The proper time is the the interval measured between events happening at the 
same point. 



Homework 4 

Item Imagine a ship and a rod observed to be moving together with velocity v. Does 
the rod measure the proper length? 



Homework 5 

A ship is observed by moving with velocity v, is measured with a rod stationary with 
respect to the observer, does the rod measure the proper length? 



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Homework 6 

A ship moving at a speed of v — 0.80c relative to a particular satelite, If the satalite 
measures a time of 2.00s for the ship to pass the satellite what is the proper time? 



Homework 7 

A ship moving at a speed of v — 0.80c travels from Earth to a star the clock on the 
ship measures a time of 90 years. What is the proper time? 



Homework 8 

How fast must a muon travel such that it's mean lifetime is 46/is, if its rest lifetime 
is 2/is. 



Homework 9 

A spaceship travels to a star 95 light years away at a speed of 2.2 x 10 8 m/s. How 
long does it take to get there (a) as measured on the earth? (b) as measured by a 
passenger on the spaceship? 



4.4.4 Time discrepancy of an event synchronized in K as 
viewed in K' separated by x^ — X\ 

The time discrepancy of an event synchronized in K viewed in K' where the events 
are separated by (x 2 — X\). 

(4.71) 
(4.72) 
(4.73) 
(4.74) 
(4.75) 

4.4.5 Time discrepancy of events separated by At and Ax in 
K as viewed by K' . 

4.5 Doppler Effect 

A Source stationary in K emits iV waves in a time At. If K moves with speed v with 
respect to frame K', while in frame K' a receiver sits stationary with respect to K' . 



Tl 


= lih- 


Vxi 
c 2 ' 


Tl 


= 7(*2" 




Tl ~ Ti 


= l{t 2 - 


tl)-^- v :') 

c z c z 


^2 — ti 


= ^(*2 


- Xi) 


At 







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Chapter 4 Introduction to Modern Physics: Physics 311 

What is the shift in frequency received in K' from that emitted in K. The wavelength 
received is calculated as, 

A' = < 4 ' 76 > 

Where At R is the time for the wave to travel from the source to the receiver in the 
receiver's rest frame. Now we calculate the received frequency as, 

h = £ = ^-_JL (4.77) 



A' c - v At R 
1 N 



Given N = fsAt s , and At 5 = At p , 



1 - £ 2 

c 



For a source approaching a receiver, 



c 



As functions of wavelength, 



A 



[r 

fs approaching ^R 
/fl ^5 



'1+ 






c 




'(' 




c 


/l- 


1) 




c 



/is receding ^R \ ^~ c 



(4.78) 



' r. 



= -L//5 (4.81) 



fn = \l\^ifs (4-82) 



/* = \lY^ffs (4-83) 



/ = t (4-84) 



(4.85) 

(4.86) 
(4.87) 



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4.5.1 Example: Doppler Effect 

Assuming a distant galaxy is moving at v — 0.64c and it gives light at a wavelength of 
Ao = 656nm, What is the new wavelength if the galaxy is approaching? or receding? 
Approaching: 



As 
Xr 



Which is a shift toward the color blue 
Receding: 



As 



which is a shift toward red. 

If most spectra of distance galaxies appear to be longer in wavelength than ex- 
pected the universe is expanding. 

4.6 Twin Paradox 

Assume pair of twins, Abraham and Bill, Bill goes on a trip at a speed of v to a 
distant star, after reaching the star Bill turns around and returns at a speed of —v. 
What are their relative ages? 

Assume the star is 4 lightyears away, and bill travels at a speed v = 0.80c. In 
Abraham's time frame, bill travels 5 years before turning around. 



While bill's clock is slower, 

At 

the return trip takes the same amount of time in Abraham's frame. So lOyears has 
passed for Abraham, while on 6 years has passed for Bill. 
DRAW A SPACE TIME DIAGRAM 




(4.88) 

307nm (4.89) 




(4.90) 
(4.91) 



L n 4 . 

— = = 5years (4.92) 

v 0.80 y v ; 



At 5y 5 , A . 

3years (4.93) 



7 rZni 5/3 



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Chapter 5 

Velocity, momentum and Energy 



Having found the point transformations, 



X 


= j(£ + vt) 


(5.1) 


t 


= 7(r+^0 
cr 


(5.2) 




= j(x — ft) 


(5.3) 


T 


, v . 
= 7(^-^) 


(5.4) 



5.1 Velocity Transformations 

How does one transform a velocity of u in frame K to a velocity of u' x in frame K', 
if K' is which is moving at v with respect to K. Clearly the velocities are given by, 



u , 



Mr = 



At 
Ax 

At 



U„ = 



A?? 

a7 

Ay 
At 



(5.5) 
(5.6) 



Then, 



A^ 



7(a; 2 - wt 2 ) - 7(^1 
7(Ax - uAi) 



(5.7) 
(5.8) 



and, 



At = 7(At - -tAx) 



(5.9) 



26 



Chapter 5 



Introduction to Modern Physics: Physics 311 



Substituting (5.8) and (5.9) into equation (5.5) gives, 



Ux ~ l(At-^Ax) (5 - 10) 

^ V 1 c 2 At ' 

(5.12) 



Substituting (5.8) and (5.9) into equation (5.5) gives, 



u y = T7T7 ( 5 - 13 ) 



7 (At - %Ax) 

1 U„ 



"y 



7 1 - Jt«:e 
1 jj^l 



Then the velocity transforms are, 



^2 



(5.14) 
(5.15) 



5.1.1 Example: transforming the velocity of an object mov- 
ing at the speed of light 

Now, let us apply this to an object moving at c in frame S'. 

u' x = c (5.18) 

u x = I f^7, (5.19) 

C + V =c (5.21) 



Homework 10 

A distant galaxy is moving away from us at a speed of 1.85 x 10 7 m/s. Calculate the 
fractional redshift ( A ~ A ° ) in the light from this galaxy. 



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Homework 11 

Two spaceships are approaching each other (a) if the speed of each is 0.6c relative to 
the earth, what is the speed relative to each other? (b) if the speed of each relative 
to the earth is 30,000m/s, what is the speed of one relative to the other? 



5.1.2 Transformation of the angle of a light ray 

Assume a light ray is traveling in frame K at and angle 9. How is this ray viewed 
from K' which is moving at a speed of v relative to K. 

u x = ccosO (5.22) 
u y = csin9 (5.23) 

Applying the velocity transforms we find the x' and y' coordinates of the light ray, 

CCOS0 + V 

u * = rn^ (5 - 24) 

c 

The angle 9' can then be calculated from either the sin or tan functions, 

tanS - = ^=( cM f^ )( 1+ j n me ) (526) 

(5.27) 



u', r V 1 + - c cos 9 Ac cos 9 + v 



sinA^l -/3 2 



sin^' = — — ^ ^- (5.28) 



cos 9 + (3 
1 + p cos 9 

5.2 Transformation of the Energy of Light Rays 

We can define the energy of a light complex per unit volume as, 

A 2 

^ (5 ' 29) 

If the volume of the light complex were the same in two reference frames, S and S' 
then the ratio of energy in the S' frame to the S frame would be, 

A' 2 

^ ( 5 - 3 °) 



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Chapter 5 Introduction to Modern Physics: Physics 311 

However, the volume is not the same, because of time and length contraction, What 
is a sphere in the S frame, 

(x - let) 2 + {y- met) 2 + (z- net) 2 = R 2 , (5.31) 

where l,m,n are direction of the wave normal cosines. This surface is an ellipsoid in 

ft - l^ f + fa - mft^ 2 ) 2 + (C - nft^f = R 2 (5.32) 
The ratio of the volume of the light complex in S' to that in S is, 



V s > V 1 ~~ 

Vs = l-fcos0 (5 - 33) 

Now we can write the ratio of the energy in the primed frame to that in the unprimed 
frame as, 



E' A' 2 S' l-^cos( 



t: a 2 s / 1 _ r* 



(5.34) 



And if = then, 



E> 

Which is similar to the ratio of the frequencies, thus we observe, that 



(5.35) 



E oc / (5.36) 

5.3 Does the Inertia of a Body depend upon its 
Energy- content? 

We found previously that the ratio energy of a plane wave of light with energy E in 
the rest frame and E' in a frame moving with speed v with respect to the rest frame, 
frame is, 

E' 1 — - cos 

Now, the source gives off plane waves with energy |L at an angle to the direction 
of v, if the source has a energy E before emission and E\ after emission of two plane 
waves in opposite directions (so that there is no change in momentum), we can write 
the conservation of energy as, 

E = E 1 + h+ l -L (5.38) 



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Similarly in the moving frame we can write the energy as, 

1 1 - - cos 6 1 1 - - cos 6 
H = H 1 + -L r^-f +^L J^J? (5.39) 

H, + —£== (5.40) 

H and E can differ only by the kinetic energy and a constant, so, 

H -E = K + C (5.41) 
H 1 -E 1 = K ± + C (5.42) 



So, 



tfo-tfi = H -E -(H 1 -E 1 ), (5.43) 
= (Ho-Hj-iEo-Ej (5.44) 

- L (5.45) 



1 - 4 



= L[—L=-\) (5.46) 



Doing an expansion of the term 



A -1/2 _ IV 2 

(?) ~ 2 c 

neglecting the higher order terms we get, 

1 L 



I ) =f + higher order terms (5-47) 

\ c J 2 c 



tf - ^ = l-^, 2 , (5.48) 
Therefore if a body gives off energy L in the from of radiation its mass diminishes by 



L_ 



Am = ^, (5.49) 



C 2 



L = Amc 2 . (5.50) 



5.4 Momentum 



If a constant force is applied to a body, under Newtonian mechanics we would expect 
the body to continue accelerating up to an infinite velocity. Special relativity, however, 



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means that c is a limiting velocity. This means the inertia (mass) of a body must 
increase. 



v — > c 
m — > oo 



(5.51) 
(5.52) 



Then the mass of a body must be a function of its speed and mass at rest. 



m f(u), 
m 



1 Uf_ 

J- o 



(5.53) 
(5.54) 



the momentum can now be written as, 



p = m u u (5.55) 
= •ymou, (5.56) 

m °J= (5.57) 



Here we take a moment to note that the momentum is dependent upon the velocity 
and not directly the relative speed between the frames. 



1. p is conserved 



2. as - — > 0, -+> mu 



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Chapter 5 Introduction to Modern Physics: Physics 311 

Now we will use p to calculate the kinetic energy, 

k — y2 Fds ' ( 5 - 58 ) 

Ju=0 

U ^ds, (5.59) 



udp, (5.60) 



d 



(_^y = ma (i-«;y» d «, ,5, 2 , 

K = um yl -J ofot, (5.63) 

brr?- 1 )- < 5 - 64 ' 



moc 2 1 



.2 



A = — — m c (5.65) 



Define, 



We can write the energy as, 



E = m c 2 (5.66) 



E = K + m c 2 , (5.67) 
= K + E , (5.68) 



ITIqC 2 



1 - % 



The work done to move a mass from rest to the final energy, 



i _ u2 

1 c 2 



Which allows us to write a relativistic mass as, 



(5.69) 



m ° c2 2 /r 7n x 

= m r c (5.70) 



m r = — ^=^= (5-71) 

i - 4 



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Chapter 6 

Four vectors and Lorentz 
invariance 



6.1 Momentum 

To preserve the concept of momentum conservation it must be shown that if momen- 
tum is conserved in one frame it is conserved in all inertial frames. We start with 
some observations about momentum. 

1. Newtonian formulation is incorrect. 

2. The relativistic form must reduce to the Newtonian from when u « c. 

3. p is a vector in the same direction as u. 

4. Ap = for a collision or explosion for any inertial frame. 

The last statement is a postulate of invariance. The quantity Ap is invariant 
under transformation. Since the Newtonian form is not invariant, we need a new 
formulation. What quantities have we studied are invariant under transformation? 
It was shown previously that the quantity describing the envelope of a light wave is 
invariant, 

12 , 12 , 12 2+12 2 , 2 , 2 2j.2 in 1 \ 

x + y + z — ct = x + y + z — ct. (6.1) 
If a vector is written, 

x li = (x,y,z,ict) (6.2) 

Then, 

x M • x M = x 2 + y 2 + z 2 - c 2 t 2 (6.3) 

And can be understood at the length, L = x fJi -x^, x^or the radius of a four dimen- 
sional hypersphere. We can see that similar to Newtonian physics the length of x^ is 
invariant. 



33 



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Once our set of coordinate are written as a four vector, we can summarize the 
Lorentz transformation as a matrix operation. 



x' 




7 












X 


y' 







1 










y 


z' 










1 







z 


id! 












7 




ict 



Or written as, 
where, 



A = 



7 








ifil 





1 














1 





ipl 








7 



(6.4) 



(6.5) 



(6.6) 



Other invariants may be formed from, 

• the sum, product, quotient, or difference of two Lorentz scalars 

• 

• the Dot product, ■ of any two four vectors. 
6.1.1 Invariant interval 

Suppose an event 1 occurs at (xi, y±, z±, icti) and event 2 occurs at (#2, 2/2, ^2,^2) ■ 
The difference, 

= x% - xi (6.7) 

is the displacement four-vector. And the invariant product of Ax M is called the 
invariant interval, 



/ = Ax^Ax^ = Ax 2 + Ay 1 + Az 2 - c 2 At 2 
= — c t + a 

Where t is the time interval between two events and d their spacial separation. 



(6.8) 
(6.9) 



1. / < the interval is time-like (events that occur at the same place but separated 
by time) 

2. / > the interval is space-like (events that occur at the same time but are 
separated by space. 



3. / = light-like events are connected by a signal traveling at the speed of light. 



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If / < then there exists a system in which the events occur at the same point. 

If / > then there exists a system in which the events occur are synchronized. 
The boost velocity v would have to be greater than c to place them at the same place. 

Since the Lorentz transformations include the time as well as space, we may treat 
time as another dimension of space. 3-Dimensions of normal space and one dimension 
of time, this is known as Minkowski-space. 

Let us consider a light ray produced at the origin and moving to the right with a 
slope of c. 

INSERT GRAPH 

Consider a light pulse expanding in all directions on a plane, after sometime t. 
The tip of the light pulse can be expressed as r = ct. Similar to the above graph if 
we plot this ever expanding circle as a function of t as well, this ever expanding circle 
forms a cone around the time axis. 

We might imagine, though we can't really draw it, A series of rays expanding in 
3-D. Now, we have an expanding sphere in time with the time axis plotted in a 4-D 
hyper-cone. 

Typically we plot ct rather than t, so that all dimensions have the same units. 

We can also extend the time axis in to the past getting a past light cone which 
represents a shrinking sphere of light (non-physical). This gives us a double cone 
defined by r 2 = c 2 t 2 . This Minkowski space represents the past, present and future, 
the light cone separates the world in regions. 

Minkowski space permits an easy way of representing all motions of a body. It 
does this by a single line know as a world line. This world line represents the potion 
of a body at any time. Since speeds of the body must be equal to or less than the 
speed of light. The angle the world line makes with the time axis will be less than 
the angle of the light cone. 

The tangent of the world line gives its velocity. 

The light cone represents all points in the future which can be reached from the 
current point, and all points in the past the could have reached the tip. 

Since simultaneity depends upon the reference frame. Two events A\ and B\ are 
simultaneous relative to frame R, they are not simultaneous in R' . Recall previously 
the diagram in which we illustrated the condition of simultaneity. The line A\B\ 
is parallel to the but that same line as seen from a moving frame. A' l B[ is 

not parallel to the rr'— axis tilted lines indicate simultaneity in a moving frame, in 
Minkowski space-time that tilted line becomes a tilted x — y — z — t space. 

The now space for a moving observer in R' is tilted with respect to the now space 
of the observer at rest in R. 

INSERT IMAGE 

Therefore every event outside the light cone is simultaneous with the origin if seen 
from a suitable frame. The slope of the line connecting two events tells you if the 
invariant interval is time like (slope > 1/c) or space-like (slope< 1/c). 



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The minus sign in the invariant interval gives rise to an important geometric result. 
Minkowski space is hyperbolic, and in 4 dimensions a hyperboloid. Under rotation 
around a spatial axis, point p describes a circle in the other two spacial coordinates, 
given by r 2 = x 2 + y 2 . Under a Lorentz rotation the interval I = d 2 — c 2 t 2 is preserved. 
The locus of all point with a given J is a hyperboloid. Time like / have a hyperboloid 
made of two sheets, while a space-like hyperboloid consists of 1 sheet. 

No, amount of transformation will carry a point from the lower sheet to the upper 
sheet of the time-like hyperboloid. This means that if the invariant interval between 
two events is time-like, the ordering is absolute; if the interval is space-like the ordering 
depends on the inertial system. The invariant interval between causally related events 
is always time-like. 



Invariant interval and the proper-time 

The invariant interval was defined as, 

I = Ax 2 tl = Ax li Ax^ = Ax 2 + Ay 2 + Az 2 - c 2 At 2 (6.10) 

= -c 2 At 2 + d 2 (6.11) 



Where a clock in our rest frame measures a movement of a/ Ax 2 + Ay 2 + Az 2 in the 
time interval At between two events. By a clock in the objects rest frame the two 
events are separated spatially by V = x' 2 — x[ = and a time At'. Since this interval 
is measured by a clock stationary with respect to the points x' 2 = x[, At' measures 
the proper time. 
Then we write, 

c 2 At /2 = c 2 At 2 p = c 2 At 2 - Ax 2 - Ay 2 - Az 2 (6.12) 

Ax 2 + Ay 2 + Az 2 ' 



At 2 = At 



p 



(6.14) 



Since, 
Leads to, 



Ax 2 + Ay 2 + Az 2 

At 2 =V (6 ' 15) 



At 2 p = At 



(l-^) (6.16) 
At p = At^l-P 2 (6.17) 

^ = At (6.18) 

7 

This shows that the invariant interval is simply the square of the proper time, 

I = -cAt 2 p (6.19) 



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6.1.2 four-velocity 

Previously we defined the velocity relative to quantities in the same frame, 

dx 



u = 



dt 



Let us define the four-velocity as, 



Ax, 



(6.20) 



(6.21) 



where At p is the proper time. The proper velocity is now measured relative to the 
time the traveler measures. Since, 



At p = 



At 

7 



We find that the four-velocity is given by, 

Ax 
= 7 At = 

c 

T] 4 = 



U 



(6.22) 

(6.23) 
(6.24) 



In frames moving with velocity V, the four-velocity transform with, 



Vx 



Vy 
Vz 



(6.25) 
(6.26) 
(6.27) 
(6.28) 



or 



~ix 




7 









Vz 














(6.29) 



7 if3-f 
10 
10 

■i/?7 7 

Transforming again with the rotation, A, in fact all four-vectors will transform using 
the matrix A. 





Vx 




% 




Vz 







6.1.3 Four- momentum 

We will define the four-momentum as, 

Pf, = rrioeta^ 



(6.30) 



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n(n - 1) , 



If we expand 7 by the binomial expansion, 

(1 -x) n = 1 -nx- '" v \ ^ x 2 + ---, (6.31) 
where u « c then, 

P = ~^L= = mu(l - %Y'I\ (6.32) 

u 2 C 



V = mu(l- \{-f + |(-) 4 + •••)• (6-33) 
2 c 8 c 



As - — >■ then, p = mu. The component of p^ expands as, 



mc 



Pa = , (6.34) 



I V? 

1 c 2 

1 U 2 

= mc+-m h • • • (6.35) 

2 c 

So that as - c — > then, the p 4 c reduces to the kinetic energy \mu 2 . Thus we write 
the kinetic energy as, 

K = mc 2 ( - 1) (6.36) 

We observe; 

1. p parallel to u 

2. p goes to the classical result as u « c. 
6.1 A Conservation of Momentum 

6.1.5 Classical momentum conservation under a Galilean trans- 
formation 

Given the classical momentum of p = mv examine a collision between two masses A 
and B, that result in two new mass C and D, then, 

m A v A + rriBVB = m c v c + m D v D . (6.37) 

Now, calculate the transformation to K' where the velocity transforms in the Galilean 
method by v ' = v — u, 

mAVA + m B VB = m c Vc + m D v D (6.38) 
m A (v' A - u) + m B (v' B - u) = m c {v' c — u) + m D (v' D — u) (6.39) 



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Collecting like terms, 

rriAv' A + rriBv' B — mcv' c — m B v D + u(m A + tu b — mc — m D) = (6.40) 

The first term gives momentum conservation if the second term m A + m B — mc — rrio 
is zero. Classically this is known as mass conservation. 

6.1.6 Relativistic Momentum Conservation 

Let us examine a collision between two masses A and B, that result in two new mass 
C and D in K' with velocities a', b', c'd' respectively, the momentum perpendicular to 
the boost is given by, 

m AVA y + rriBVB y = m cVc y + m cv' Dy (6-41) 

Since the transform of a 4-velocity perpendicular to the boost is given by, r\ y = r)' y , 
then, 

m A r]A y + m B r] By = m c r]c y + r m c r] Dy (6.42) 

Therefore momentum is conserved in the y, z directions if the boost is in the x direc- 
tion. 

In the direction parallel to the boost we have transforms, 

V'l = l(Vi (6.43) 
Va = l(VA-M (6-44) 

The momentum in the parallel direction is given by (dropping the indice 1), 

m A ri' A + m B r}' B = m c r( c + m c r]' D (6.45) 
m A (l(r]A ~ Pvm)) + m B{l{i]B - PvbJ) = (6.46) 
mc(l(Vc ~ Pvc 4 )) + ™d(i(vd ~ PvdJ) (6.47) 

Collecting like terms we arrive at, 

m A r] A + m B 7] B - m c r)c - m D r) C + f3(m A r] A4 + m B r] B4 - m c r] C4 + m D r] D4 ) = 0(6.48) 

The first term like in the classical solution is simply the conservation of momentum 
if, the second term is zero, 

m A r] M + m B ri B4 - m c r]c 4 + r m D r] D4 (6.49) 

We observe, 

N 

E ^= = ° < 6 - M > 

„2 



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is conserved. Multiplied by c gives, 

N 

E 



rrijC 



1 - % 



(6.51) 



In equation (6.36) the relationship to kinetic energy was shown giving equation (6.51) 
the meaning of conservation of energy, We now define the energy as, 



and the mass as, 



The quantity, 



E 



M 



mc 



1 - 



m 



1 uf_ 

-L 9 



E = Mc 2 



(6.52) 



(6.53) 



(6.54) 



is conserved. m is sometimes called the rest energy and E = m c 2 as the rest energy. 
From the relations, 



P 



E 



mu 



mc 



It can be shown, 

pc = uE. 

And the four-momentum can be written as, 



= (Pl,P2,P3, 



(6.55) 
(6.56) 

(6.57) 
(6.58) 



Which transforms using A, 



Vi" 




7 












~P1 


P'2 







1 










P2 


P's 










1 







P3 


I* 
L c J 












7 . 




L c 



(6.59) 



or 



Pi c — 1Pi c — PlE 

p' 2 c = p' 2 c 

p' 3 c = p' 3 c 
E' = -fimc + 7# 



(6.60) 



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Since p^ is a four-vector, we construct an invariant for the momentum 

E 

P>m = P-p-(—) 2 



p ■ p 



,E 2 TJIqU ■ U ITLqC 2 



V 2 



, 1 



m^c 2 ,u 2 



l-4 K c 2 

2 2 

— m c 



-1) 



p-p-{^) 2 = -m 2 c 2 (6.61) 
or as it is more commonly written, 

E 2 - p 2 c 2 = m 2 c 4 (6.62) 

6.1.7 Particles of zero mass 

Given, 

pc = ; (6.63) 

i y?_ 

mc 2 



E = (6.64) 



c z 



pc = uE (6.65) 
E 2 -p 2 c 2 = m 2 c 4 (6.66) 

We can see that as " — > 1, the mass has to vanish, leading to to relationship between 
energy and momentum for a mass-less particle as, 

E = pc (6.67) 



Homework 12 

How much rest mass must be converted to energy to (a) produce 1J (b) to keep a 
100- W light bulb burning for 10 years. 



Homework 13 

(2.67 in Tipler) The sun radiates energy at the rate of 4 x 10 26 W. Assume that this 
energy is produced by a reaction whose net result is the fusion of 4 H nuclei to form 
1 He nucleus, with the release of 25 MeV for each He nucleus formed. Calculate the 
sun's loss of rest mass per day. 



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Homework 14 

(2.70 in Tipler) Show that 



<yi=P7p)= m »( 1 -'" 2 / c2 )" /2 *' < 6 ' 68 > 



Homework 15 

(2.79 in Tipler) For the special case of a particle moving with speed u along the y 
axis in the frame S, show that its momentum and energy in frame S' are related to 
its momentum and energy in S by the transformation equations 



p' x = Pfa-^) (6-69) 

Py = Py (6-70) 

p' z = p z (6.71) 

& = V Px 

c V c 



= 0(B-YPe) ( 6 . 72 ) 
V c c / 



6.2 Mechanical Laws 

We can see that this formulation of the momentum poses a difficulty for Newton's 
laws of mechanics. Let us examine Newton's second law which we will write as, 

*=£ (6,3) 

In terms of the force the work is given by, 

W — J F-dl (6.74) 

Now combining we get, 

w = i ^. d i= i d 4-€dt= ! d 4-udt 

udt 



dt J dt dt J dt 

d f mu \ mu du d / mc 2 \ dE 



/ mu \ 7 



dt dt\ L _ y?J (i_^)3/2 dt dt\ L _ v?J dt 



dE 

W = I —dt = E fina i - E initia i (6.75) 



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6.2.1 Motion under a constant force 

Imagine a particle of mass m that is subjected to a constant force F. In classical 
mechanics we imagine that this implies a constant acceleration leading to an infinite 
velocity. Since this is a clear violation of the second principle of special relativity, we 
reexamine this problem in terms of the relativistic momentum. Then, solving for x, 
as a function of time. 

dp 



dt 



= F 



(6.76) 



implies, 



Ft + constant (6.77) 
If the particle starts from rest then at time t — 0, p — and the constant is zero. 

mu 



p 



Ft 



(6.78) 



solving for u, we arrive at, 



u 



(F/m)t 

Ji - {^y 



(6.79) 



integrate again, 



x 



F 

m 



a 



-.dt' 



mc 



■Ft'\2 
> mc ' 



i - (— ) 2 IS 

mc 



mc 



Ft' 

— ) 2 -l 

mc 



(6.80) 



Classically the solution is a parabola x = [Fj1m)t 2 this solution is a hyperbola. 
As a constant force is applied the velocity of the particle asymptotically approaches 



c. 



6.2.2 Cyclotron Motion 

The trajectory of a charged particle in a uniform magnetic field is circular or cyclotron 
motion. The magnetic force is given by, 



F = quB 

In special relativity we need to calculate the centripetal force. 

dp d9 u 
F — — — p— = p— 
dt F dt F R 



(6.81) 



(6.82) 



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now, 



quB = p^- (6.83) 
R 

p = qBR (6.84) 



Homework 16 

Show that for uniform circular motion dp = pd6 



6.3 Proper Force 

It would be useful to produce a force that is a four-vector, we take as the proper force 
the variation of the, of the momentum with respect to the proper time. You should 
recall that is this a similiar definition to the proper velocity, 

A/ 

K " = w p < 6 - 85 > 



Then, 

And the 4th component is, 



K = jF (6.86) 



A' 4 =^ = ~ (6-87) 
At p c dt v ' 



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Chapter 7 
General Relativity 



Einstein cites 2 reasons for a new theory of gravitation, 

1. Special Relativity Applies to all forces but gravity, Gravity is not invarient 
under special relativity 

2. To construct an inertial reference frame (IRF), one must eliminate all forces, 
but gravity cannot be eliminated 

Falling can eliminate the force of gravity, if all other forces are also eliminated the 
falling person becomes an inertial observer. However, now the world appears to be 
in an accelerated frame. We can check the law of inertia in free fall (in a small lab 
over a short distance) the objects appear at rest or in constant relative motion and 
we seem to be moving in a straight line, in agreement with the law of inertia. 

If the law of inertia is true in free fall, and in systems under the acceleration of 
gravity, we conclude the law of inertia hold relative to an accelerated frame. 

7.1 Absolute Acceleration? 

It is then natural to ask if acceleration is absolute or relative? 

• Newton argued for absolute acceleration (2 buckets) 

• Ernst Mach Argued that if the universe had only these 2 pails they would be 
no way to distinquish between the pails and symmetry would demand the same 
sufrace on the water of both pails. 

Einstein surmized that acceration should be relative just like velocity and position. 

7.2 2 principles of General Relativity 

• Principle of Covarience 



45 



Chapter 7 Introduction to Modern Physics: Physics 311 

• Principle of Equivalence 

All Physical laws must be expressed in a form covarient with respect to an arbi- 
trary coordinate transformation. 

7.2.1 Principle of Covarience 

Under special relativity inertial reference frames are preferable due to the simplicity 
of the laws of nature as expressed in inertial frames. 

• We would like a relativity that is general in any frame. 

• We expect the laws of nature to be the same in and out of an inertial frame. 

This generaliztion destroys the uniqueness of the concept of space and time mea- 
surements. 

7.2.2 Results of a transforming rotation in S.R. 

K (X, Y, Z, T) and K rotating at u with respect to K Q . Upon transformation of 
K — > K Z remains unchanged, in K , 

X = RcosO, (7.1) 
Y = RsinO, (7.2) 



while in K we write, 



x = r cos(6 l — ujT) (7.3) 
y = r sin(6> — uiT) (7.4) 

(7.5) 



r = R because it is orthognal to u. 
The length of an arc transforms as, 



rdO 

Rd ^ _<^ == 



1 - 



The measure of the line element connecting (r, 6) to (r + dr, 9 + d6) in K is written 

as, 

r 2 df) 

dl 2 = dr 2 + y2 (7.7) 

c 2 

If dr = and d6 = the ratio of the circumference of a circle to its radius becomes, 

circumference 2n 



(rui) 2 



> 2tt (7.8) 



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We recognize that a measuring rod is I — 1/(3 in length in K and so it measures a 
larger circumference by a factor of (5. 

[INCLUDE SURFACE PICTURES] 

There are several conclutions we can draw; 

• For an observer in K space is not homogeneous or isotropic. 

• The measure of time is effected. 

• The transfer from an inertial fram to a non-inertial frame results in a complete 
loss of objectivity in spacial and temporal measurements. 

Thus we would prefer to apply relativity across all transformations. 
7.2.3 Principle of Equivalence 

No dynamical experiment can distinquish between the so-called ficticious (inertial) 
force arising in system K\ and the so-called real (gravitational) force prevailing in 
system K 2 . 

Gravity is detected when a reference frame is not freely falling. We experience 
gravity only because the earth prevents us from falling toward its center, gravity is a 
consequence of the choice of reference frame. 

This leads to the concept that the inertial mass of a body is equal to its gravita- 
tional mass. 

Eovtos measured the force of gravity on a body suspended, and the force of cen- 
terfuge arrising from the rotation of the earth. He found the inertial mass and the 
gravitational mass to be the same to 5 parts in 10 9 . Further experiments have refined 
this to 1 part in 10 11 . 

The conclusion is that inertial mass is of gravitational origin, and leads to the 
bending of light by gravity. 

If we cannot tell the difference between a frame under the acceration of gravity and 
one being accelerated by a rocket. We examine the path of a light ray in such a frame. 
In a rocket if it is accelerated, a light ray emmited perpendicular to the acceleration, 
will appear on the wall slightly lower than height at which it was emitted, because 
of the movement of the frame. If we cannot tell the difference, then in a frame 
(stationary) under the acceleration of gravity, we would expect the light to strike the 
wall slightly lower than emitted, meaning that the light ray bent slightly. 

The first observation of such a bending of light rays occured in 1919 (5 years after 
the publication of General relativity). Observervations of a star on the edge of an 
eclipse and the same star when the sun was far away from the star. This position of 
the star showed a shift. Shifts as predicted by gravitational theories, 

• Newton's Law predicts no bending of the light 



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• Newton's Law + special relativity - predicts bending exactly 1/2 the amount 
predicted by general relativity. 

• GR predicts bending (for the particular star 1.75") 

The results were about 2.0 to 1.6 arc seconds with and uncertainty of ±0.3". The 
observation is consistant only with the theory of GR. More-recently we have other 
gravity lens observations. 

Remember that in SR the velocity of light is a constant, meaning constant speed 
and direction. If a light ray is bent how can this be? 

How is a straight line defined? 

If we define a straight line as the shortest distance between 2 points, then picture 
the surface of a sphere, the shortest distance then follows a greatest circle. In general 
this is known as a geodesic. 

This light follows the geodesic on the surface of space-time. 

We now picture the sun not as a sphere but as a depression in space. 

7.3 Experimental support for General Relativity 

• Gravitational Red Shift - increased gravitational pull slows a clock 

• Comparison of Clock at Boulder moves faster than the one at the Royal Green- 
wich Observatory 

• Precession of the orbits of planets- 

- If we calculate the orbits of the planets using the theory of general relativ- 
ity, we find a precession in the orbits of the perihelion that is not present 
in the calculations based upon Newtonian gravitation. 

- Observed on Mercury (1845) 43.0 arc sec per century « 1% 

— Venus 8.63 arc sec per century 

— earth 3.8 arc sec per century 



Bodies emit gravitational radiation - a vibration in the curvature of space-time pro- 
pogation with the speed of light 

Is there evidence of such radiation? Bianary Stars, pulsars 

Black Hole a region in space with a gravitational pull larger than a critical value, 



7.3.1 Black Holes 



v, 



escape 



> C 



(7.9) 



All radiation is absorbed, and none is emitted. 



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Chapter 7 Introduction to Modern Physics: Physics 311 

In Newtonian mechanics we would calculate the escape velocity as, 



^ (7,0) 

Then if we set the escape speed equal to the speed of light and solve for the radius, 
we obtain the critical radius Rs, called the Schwarzchild radius, 

2GM 

Rs=—^- (7.11) 

cr 

For an object of the mass of our sun its radius would have to be about 3km. 



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Part III 
Origins of Quantum Mechanics 



50 



Chapter 8 

The Quantum Hypothesis 



We have seen how the rules of classical physics break down when objects move at 
speeds comparable to the speed of light. The laws of classical physics similarly break 
down when they are applied to microscopic systems (Atomic sized). 

This has lead to the Theory of Quantum Mechanics, which is the only extant 
theory that adequately describes such phenomena. 

Quantum mechanics was primarily hashed out between (1881-1932). The origin 
of quantum mechanics are in thermodynamics. 

Some Important Dates 

• 1895 Rongen discovers x-rays 

• 1896 Becquerl discovers nuclear radiation 

• 1897 J.J. Thompson discovers the electron and measures e/m, shows the elec- 
tron is part of an atom 

• 1900 Plank explains blackbody radiation through energy quantization and a 
new constant h 

• 1905 Einstein explains the photoelectric effect 

• 1907 Einstein applies energy quantization to temperature dependence of heat 
capacities 

• 1908 Rydberg and Ritz generalize Balmer's formulas 

• 1909 Millikin's oil drop experiment 

• 1911 Rutherford's discovery of the nucleus 

• 1913 Bohr Model of Hydrogen atom 

• 1914 Mosely Analyzes x-ray spectra using Bohr model 

51 



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• 1914 Frank and Hertz Demonstrate atomic energy quantization 

• 1916 Millikin verifies Einstein's photoelectric equations 

• 1923 Compton explains x-ray scattering 

• 1924 de Broglie proposes electron waves of A = h/p 

• 1925 Shrodinger develops mathematics of electron wave mechanics 

• 1925 Heisenberg invents matrix mechanics 

• 1925 Pauli exclusion principle 

• 1927 Heisenburg uncertainty principle 

• 1928 Gamow and Condon and Gurney Apply Quantum Mechanics to explain 
alpha-decay lifetimes 

• 1928 Dirac proposes relativistic quantum mechanics and predicts the positron 

• 1932 Chadwick discovers the neutron 

• 1932 Anderson discovers the positron 

We know from experience that an object that is heated gives off radiation. This 
is known as thermal radiation. Usually this radiation is in the IR range. 

• lOOOif ~ Red hot 

• 2000if ~ Yellowish-white 

The actual light is a collection of the wavelength's given off, a continuous spectrum 
with a dominate wavelength. 
[INSERT PICTURE] 

We can simulate this with the use of a Blackbody, as we recall a blackbody is an 
ideal absorber/emitter of radiation. We can think of a black body a cavity with a 
small hole, radiation that falls on the hole enters the cavity, but does not leave. 

As the temperature increases the dominate wavelength sifts, this is known as 
Wein's displacement law, 

\ max T = 2.90 x lO^mK (8.1) 

Thermal radiation results from the oscillations of atoms near the surface, clas- 
sically these oscillations are not restricted and may take any value. The classical 
calculations predict an intensity proportional to, 

I « ^ (8.2) 



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This theory is good at large wavelengths, but we can see that as, A — > / — > oo Then 
an infinite amount of energy is being radiated at any temperature. It also deviates 
from the experimental value. This is known as the ultraviolet catastrophe 

Max Plank (German 1858-1947) formulated an idea bases on the mathematical 
conjecture that these oscillators are limited to certain energies given by, 

E n = nhf n = 1,2,3, ••• (8.3) 

where / is the frequency of the oscillator. The oscillator then emits energy when it 
changes from one state to another. Since the energy comes in discrete amounts this is 
called a quantum of energy. The quantity h is a constant known as Plank's constant. 

h = 6.63 x 1(T 34 Js (8.4) 

The results of Plank's theory matches the experimental values, and solves the 
issue of the UV-catastrophe. Plank was not satisfied though, believing that his math- 
ematical trick, did not have physical significance. History would prove Plank wrong 
on the physical significance and Plank won the Nobel Prize in 1918. 



8.1 The energy density 

The total energy density of a system of oscillators may be calculated from the density 
of standing waves in a cavity, G(u)du, over the interval v to v + du, and the average 
energy per mode, U, 

u{v)dv = G{v)Udv (8.5) 
The density of standing waves in a cavity may be calculated as, 

G{y)dv = ^^du (8.6) 

The classical result may be derived from the equipartition theorem, where the 
average energy for each degree of freedom for a gas at temperature T is 1/2/c^T. A 
one dimensional harmonic oscillator has 2 degrees of freedom, one corresponding to 
the kinetic energy and one to the potential energy. Thus, the classical average for a 
standing wave is, 

U = k B T (8.7) 
and we can write the energy per unit volume as, 

u v dv = G{y)k B Tdv = ^ V ^ bT dv (8.8) 

c 6 

This is known as they Rayleigh- Jeans Formula and may be written in terms of wave- 
length as, 

U\d\ = 8 ^^d\ (8.9) 



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the energy per unit volume is given by, 

poo 

= / U\d\ (8.10) 
Jo 

d\ (8.11) 



U 

h 

nQO 87T 2 k B T 



f 

Jo 



A 4 

Clearly U — > oo as A — > 

8.1.1 number density of modes in a cavity 

On a one dimensional string of length L, the wave equation, and its solution, may be 
written as, 

d 2 1 d 2 

f = -,^r, (8-12) 



dx 2 v 2 dt 2 

f = A smut cos kx (8.13) 

In order to satisfy the boundary conditions of /(0) = and f(L) = 0, 

k = ^x (8.14) 

where, 

2L 

n = — , where n — 0, 1, 2, 3, • • • (8.15) 
A 

If we generalize this result to a three dimensional cavity of volume V = L 3 , where, 
the standing waves are electro-magnetic modes, then, 

E x = E 0x smuut cos(— — x) sm(— — z) sm(— — z) (8.16) 

Lj Lj Lj 

7-1 7-1 l U X^ / n Z 7! ' /"^Z 71 " 

E x = E 0y smu>tsm{——x)cos{——z)sm(——z) (°-l7) 

Lj Lj Lj 

E x = E Qz smuit cos(— — x) sm(— — z) cos(— — z) (8.18) 

Lj Lj Lj 



Then n = (n x ,n y ,n z ), and 



n 



-n 2 x + n 2 y + n 2 z = { — ) 2 (8.19) 

To find the number of wavelengths that excede the given value of A m , 

4L 2 

n 2 x + n 2 y + n 2 z <—, (8.20) 



2L 



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The point (n x ,n y ,n z ) is on the surface of a sphere of radius to calculate the 
number points contained in a volume of radius |^ we simply integrate over the area 
of a sphere for with a radius between n and n + dn, 



, 2L 

N 



= [ A Aim' 2 dn' (8.21) 
Jo 

N = ^(f ) (8.22) 

We are only concerned with the density of states in the positive octant, dividing by 
8, and allowing for two transverse polarizations per wave, we obtain, 

where n is the volumetric number density. And correspondingly the number of points 
between A and A + AA, 

dn = ^d\ (8.24) 

Writing in terms of the frequency, A = c/u and d\ = cjv 2 dv, The number density of 
modes in a cavity is, 

G(X)dX = ^d\ (8.25) 
v 2 

G{y)dv = 8tt— dv (8.26) 



8.1.2 Solution by Micro-canonical Ensemble 

Assume that the energy of a quantum mechanical oscillator is, 

E = se = shu, where s = 0, 1, 2, 3 . . . (8.27) 

This means than an oscillator emits radiation at a frequency of v when it drops from 
one state to the next lower one. Each energy bundle of hv is known as a quantum of 
energy. And the total energy may be written in terms of the average energy of each 
oscillator, U, 

U N = NU (8.28) 

Un is a discrete quantity of an integral number of finite equal parts, each such part 
having an energy element e. The black-body is made of N oscillators (with energy 
e = his, with P energy elements distributed among the oscillators. 

U N = Pe (8.29) 



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And by extension, 

P U , 

n - 7 < 8 ' 30 > 

Label the resonators n = 1,2,3, •• • ,N, and place P energy elements in the N 
oscillators. For example, one such complex for N = 10 and P = 100 would be, 
123456789 10 
4 28 11 9 22 4 5 5 12 

The number T of all possible permutations (A perumation is different if the same 
numbers are in different locations) is given by, 

N(N + 1)(N + 2)...(N + P-1) (N + P-l)\ 
T - 1-2-3---P " NlP\ (8 - 31) 

The probability, W that a system of N resonators has energy Un is proportional to T 
the possible permutations of all distributions of P among N resonators. The entropy 
of a system is given by, 

S N = k log W + const. (8.32) 
Which can be related to average entropy of one oscillator, S, as, 

S N = NS (8.33) 

And writing in terms of T, 

/(N + P - IV \ 

S N = fclogr = Hog( l ^, p! } ) (8.34) 
Applying Stirling's Approximation to the highest order, 

N\ = N N (8.35) 

T can be written as, 

And the entropy is written as, 

S N = k{(N + P)\og(N + P) - N log N - P\ogP} (8.37) 
Writing S N in terms of P/N we get, 

S N = kN{(l + ^) log(l + - £ logPiV} (8.38) 
We have from equations (8.33) and (8.30), 

S = k{(l + j) log(l + 7) - 7 log ^} (8.39) 



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A suitable definition of the temperature, T, is given from, 

1 _ dS 
f~dU 

Differentiating S we arrive at, 



If we define (3 = then we write, 



U' 

U 



(8.40) 



^ = M^log(l + ^)- Jlog(^)} (8.41) 



h = log(l + |) - log(^) (8.42) 
= log(l + £) (8- 4 3) 



= 1 + ^ (8.44) 



If the energy e is a function of the frequency, u, 

e = hv (8.46) 

Then average energy, U, is given by, 

u = J^r- ( 8 - 47 > 

and the energy density u v can be written from the density of states, 

8nv 2 TT 8irv 2 hv . . 

gives Plank's Radiation Law. 

8.1.3 Solution by canonical ensemble 

The probability that a system is in energy state Eg is proportional to, 

p s = e~ pEs (8.49) 
where (3 = -^pf This means that after normalization we can write the probability as, 

e -PE s 

^ = (8.50) 

e -0E s 



(8.51) 

Z = J2 e ~" Es (8-52) 



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Where is a Z quantity known as the canonical ensemble. The average energy is then, 

U=(e) = , (8.53) 

and the entropy as, 

S = ~(-k B T\nZ) (8.54) 
For continuous energies the sums become integrals, 

exp[-/3ff s ] 

Ps = J-eM-PE}dE (8 - 55) 
hEexp{-/3E} , x 

If the energy is taken as the classical energy kT then we see that U quickly goes to 
infinity. 

However if we take then energies as discrete, having values of, 

E s = se, where s = 0, 1, 2, 3, • • • (8.57) 
e = hv (8.58) 

s is a number describing the number of photons in a cavity. The mode may be excited 
only in units of energy proportional to hv. (omitting the zero point energy l/2hv.) 

Similar to the quantum mechanical oscillator with frequency v, the energy eigen- 
values are integral multiples of hv. Where s is understood as the number of photons 
contained in a mode, (in the QM oscillator s would be the quantum number.) 

The quantity Z may be written more simply if the sum is evaluated, 

z = e ~ finhv = 1 + e ~ phu + e ~ Whu + ■■■ ( 8 - 59 ) 

n 

= (8-60) 
similarly we would like to evalute the sum in the average energy, 

(e) = (8.61) 

= hv^^- (8.62) 

Zj 

now we evaluate the sum as a derivative of the partition function Z. 



dq ^ ^ dq 

d 1 -e~ q 
dql-e-i (l-e-') 



(8.64) 



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Combining eqns (8.26,8.62) we arrive at, 

< e > = Mo^pHi^r (^5) 

_ e -/3hu 

u = U = J^i ( 8 - 67 ) 

Combine equations (8.5,8.26,8.67) to write the energy density as, 

. . , 87ihu 3 1 . 

u(u)d u = c3 ePhv _ 1 du, (8.68) 

= ^^TI^. (8.69) 

Plank calculated values of, /i = 6.55 x 10~ 27 erg/sec and k B = 1.346 x 10~ 16 erg/K, 
while the standard values today are h = 6.626 x 1(T 34 J/s and k B = 1.38 x 10~ 23 J/K 

The results of Plank's theory matches the experimental values, and solves the 
issue of the uv-catastrophe. Plank was not satisfied though, believing that his math- 
ematical trick, did not have physical significance. History would prove Plank wrong 
on the physical significance and Plank won the Nobel Prize in 1918. 



8.2 Einstein and the Photoelectric Effect 

Einstein used Plank's concept of energy quantization to explain the photoelectric 
effect. He claimed that energy quantization is a fundamental property of E-M radia- 
tion. 

If an Atom moves from E\ = n-Jif to energy E 2 = ^hf, then the energy emitted 
by the atom will have an energy of, 

E 2 - E 1 (8.70) 

Einstein stated that this energy has one state, 

E 2 -E l = hv (8.71) 

where v is the frequency of the Electromagnetic radiation emitted. This is a quantized 
packet of energy, he named a photon. Each photon has an energy defined by, 

E = hv. (8.72) 

We notice 2 things; 

1. Light being in discrete quanta implies that it acts like a particle. The intensity 
oc Number of photons. 

2. Light has a frequency implying wave behavior, interference and diffraction. 



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8.2.1 The Photoelectric Effect 

Certain Metallic materials are photosensitive, when light strikes the surface electrons 
are emitted. 



Incoming red light 



collector plate 




electrons don't get to plate 




Ammeter 



no current flows 



If these electrons are placed in an electric potential then the electrons will move 
creating a current. 

When the photocell is illuminated; 

• Postive voltages the current does not vary w/ voltage only w/ intensity. Ip oc 
Intensity. 

• V < retarding voltage, electrons released from "cathode" must have enough 
K to over come the retarding voltage. 

• Only electrons with K > eV can produce a photo-current. 

• At V (stopping potential) Ip — > 



K = eV 



(8.73) 



V Q same for all intensities 

No emissions for light / < f t , f t is the cutoff frequency. 



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Photocurrent vs Cell voltage 




j Saturation current 




. reference beam 


< 


f lower frequency same intensity 


E / 

u / 

£ /J 

Jz / / 
Ql / I j 


r / lower intensity same frequency 


stopping voltage / / 





-S -2 -101134ft 



phstot*!! voltage M 




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If light is in photons then the electron receives its energy from a photon. Intensity 
is the number of photons, but the energy absorbed by each electron is unchanged. One 
photon only has the energy to dislodge one electron, at a certain energy. We call the 
work function which is defined as the energy necessary to remove an electron from the 
cathode. The energy of the electrons emitted are given by Einstein's photo-electric 
equation, 

{^mv 2 ) max = hf - 4> = eV (8.74) 

Where Vo is the stopping potential. 

We can see that if the stopping potential goes to 0. there is a minimum frequency 
that will eject an electron. This is called the threshold frequency, f t . 

he 

<f> = hf t = - (8.75) 
M 

similarly we can define a threshold wavelength X t . 



Homework 17 

Photoelectric effect experiment 

1. the apparatus for the h/e measurement is already set up by your instructor. 
Do not adjust. 

2. Turn on the Hg light Source. 

3. Turn on the h/e apparatus (photoelectric effect) 

4. connect the voltmeter to the h/e appartatus. 

5. Turn on the voltmeter. 

6. Slowly rotate the h/e apparatus until the yellow light shines on the slot of the 
apparatus. 

7. Only one color should fall on the photodiode window. Do not use the 
"RELATIVE TRANSMISSION" slide. 

8. Place a yellow filter on the surface of the white reflective mask on the h/e 
apparatus. 

9. Zero the h/e apparatus. 

10. Read and record the output voltage on the voltmeter. This is the stopping 
voltage. 

11. Repeat for each color, be sure to use the appropriate filter. 



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12. Plot V vs f, and find a slope, record values of the fit. 

13. the vertical intercept is proportional to the work function <f> = eV Q , where V Q is 
the vertical intercept and e = 1.6 x 10~ 19 C. 

14. Compare <fi to the correct value of 2.32 x 1CT 19 J. 

15. The horizontal intercept gives the cutoff frequency, find the cutoff frequency f 
and compare to the correct value of 3.5 x \O lA Hz. 

16. The slope gives the ratio h/e compare to the correct value of 
0.414 x 1(T 14 Js/C 



Example: Photon Energy 

Calculate the photon energy for light of A = AOOnm and A = 700nm. he = 12A0eVnm 

E = hf = ^ (8.76) 

E(X = 400nm) = 3.1eV (8.77) 
E(X = 700nm) = 1.77 eV (8.78) 

Example 

The threshold wavelength for potassium is 546nm (a) what is the work function of 
potassium? (b) What is the stopping potential when light of A = 400nm is incident 
on potassium? 
SOLUTION: 

* = = ^-.20e, (8,9) 

(b) 

{\mv 2 ) max = hf-<j> = eV (8.80) 
eV = hf - <f> = S.lOeV - 2.20eV = 0.90eV (8.81) 

Because the term 'e' or one electron appears on both sides of the equation, it may be 
canceled and the result is 

V = 0.90V (8.82) 

Einstein's model of light is consistent with all of the experimental results of the 
photoelectric effect, a phenomena not previously explained. 



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Homework 18 

(2.1 in Tipler) Find the photon energy in joules and in electron volts for an electro- 
magnetic wave inf the FM radio band of frequency 100MHz. 



Homework 19 

(2.7 in Tipler) The work function for tungsten is 4.58eV. (a) Find the threshold 
frequency and wavelength for the photoelectric effect. Find the stopping potential if 
the wavelength of incident light is (b) 200 nm and (c) 250 nm. 



Homework 20 

(2.9 in Tipler) The threshold wavelength for the photoelectric effect for silver is 262 
nm. (a) Find the work function for silver, (b) Find the stopping potential if the 
incident radiation has a wavelength of 175 nm. 



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Chapter 9 

Quantum mechanics II: The Atom 



9.1 The Electron 

• 1858 J. Pliicker: for pressures less than P < I0~ e atm, emits invisible rays that 
carry charge (only visible when they strike the walls) These are termed cathode 
rays. 

• 1879 William Crookes: showed 

— Cathode rays travel in straight lines and carry momentum 

— Bent my magnetic fields 

— negatively charged 

J.J. Thomson's paper on cathode ray addresses the controversy of charge carriers 
through experimentation. There were two possibilities for the make up of cathode 
rays. 

1. Cathode rays: are "due to some process in the aether" ie a wave. 

2. Cathode rays: are electrified particles: rays are material, and the ray is the 
path of some charged particle 



Homework 21 

What experiment (s) could be performed that would provoe or disprove these two 
conjectures? 

9.1.1 J.J. Thomson discovers the electron 

Thomson discusses 5 experiments: 4 performed by him and 1 that was previous to 
the paper. 



65 



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Perrin's Experiment 

Two cocentric cylinders separated by an insulator are placed in-front of a cathode 
ray. There is a hole in each cylinder so that the cathode rays can reach the inner 
cylinder. 

• when the ray is not deflected by a magnetic field, the inner cylinder picks up a 
negative charge. 

• when he ray is deflected there is no charging on the inner cylinder. 

"This experiments proves that something charged with negative electricity is shot 
off of the cathode, ...and that this something is deflected by a magnet; it is open, 
however, to the objection that it does not prove that the cause of the electrification 
in the electroscope has anything to do with the cathode rays." 

Thomson 1: Perrin Thomson's way 

Now the apparatus has the same concentric cylinders but the are arranged such that 
the inner cylinder is only charged if the cathode ray is deflected such that it strikes 
the holes. Without deflection the ray does not hit the cylinders. 

• When the cathode rays do not fall on the slit, there is no charging. 

• When the rays hit the hole the charge changes 

The conclusion is that the negative electrification follows the path of the cathode 
rays. 

Thomson 2: Deflection by electric field 

There is (was) an objection to the idea that cathode rays are a charged particle, in 
that under small electric fields there was no deflection of the cathode ray. 

Thomson repeated the experiment showing the same result, but he doesn't stop 
there. He continues to examine this in more detail, proving that the absence of 
deflection is due to the conductivity passed to the gas by the cathode rays. 

Observations 

1. When the experiment is performed under a vacuum there is a deflection (after a 
time the deflection goes away due to the small amount of gas still present being 
turned into a conductor) 

2. under low pressure and high potential, the cathode ray is deflected, when the 
medium breaks down(electric discharge), the ray jumps back to its undeflected 
position. 

3. When the rays are deflected by an electrostatic field the phosphorescent band 
breaks into several bright bands separated by dark spaces. This is analogous to 
Birkland's magnetic spectra. 



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Thomson 3 

This experiment uses the same apparatus as the previous, but it is used to examine 
the conductivity of the gas enclosed in the cathode tube. Then charge flow and 
deflection can be measured to determine some properties of the gas enclosed in the 
tube. 

Thomson 4: magnetic deflection of the cathode rays in different gases 

Thomson places a cathode tube between two Helmholtz coils this allows him to mea- 
sure the radius p, of curvature on the cathode ray as related to the magnetic field 
H. 

Radius of curvature and m/e 

we can related the momentum of the charged particle to the field and the radius of 
curvature as, 

™=Hp = I (9.1) 

e 

The total charge that is transferred is, 

Ne = Q (9.2) 
While the work done on the cathode is 

W = ^Nmv 2 (9.3) 

now we can write, 

27" = Q (9 ' 4 > 

and, 

™ = ^ (9.5) 

W can be measured from the deflection of a galvanometer if the ray strikes a metal 
of known specific heat. This allowed Thomson measured m/e ~ 0.5 x 10~ 7 . 
Actual m/e = 0.569 x 10~ 7 emu/ gram = 5.68 x 10~ 12 kg/C 



Homework 22 

PART I: m/e 

I. Connect the poles of the — 24 Volts DC output to the Helmholtz coils, select 
current on the meter select (so we can read the current off of the upper scale 
of the power supply). Position the DC Voltage and the DC current knobs 
toward the middle. Keep the current near 1A, do not exceed 2A. 



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2. Connect the high voltage power supply to the e/m apparatus by, setting the 
knob for the 3A max on the "6" and connecting its two terminals to the 
heater on the e/m apparatus. Set voltage to 500V, connect the 0V and 500V 
(red high voltage pole) to the terminals on the electrods on the e/m 
apparatus. Adjust the 500V knob until the voltmeter reads a voltage between 
100-120V. Do not excede 300. 

3. Make sure the current adj knob on the e/m apparatus is not off. 

4. Set the ammeter on 4A DC and place in the circuit with the low voltage 
supply. 

5. Connect the voltmeter to the e/m apparatus, set the voltmeter to 400 V DC. 

6. Have this all checked. 

7. After the equipment is turned on, record, I, V, and measure and record the 
diameter of the circular path. 

8. Calculate the magnetic field by, B = 7.80 x 10~ 4 I 

9. Calculate e/m from equation (??) 

10. Calculate the accepted value for e/m from e = 1.6 x 10~ 19 C and 
m = 9.11 x 10~ 31 kg, compare this to the measured value. 

Thomson 5: m/e with E and B fields 

For a electron passing through a constant magnetic field, we can calculate the velocity 
from the deflection, 

F = qE (9.6) 

v z = a z t = 1 (9.7) 

m e 

t = — (9.8) 

V x 

~eE I 

v z = a z t= 9.9 

m e v x 

tanfl = V _± = _^L (9.io) 

v x m e v l x 

Note: the v m 0.1c 



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Now if we add a transverse magnetic field we can select a velocity that passes 
through the combined E, and B fields without deflection, 

F z = -eE + ev x B y = (9.11) 

E (9-12) 

(9.13) 



B 



Now from the result previously, 



tan £ = V _± = _^L (914) 

v x m e vi 

e E 

— = -T^tanfl (9.15) 
m BH y ' 

Thomson's Conclusions 

1. The charge carriers are the same for all gases 

2. The mean free path of the charge carriers depend on the density of the gas. 

3. Atoms are different aggregates of the same particles of which the electron is one 

4. An Electric field is sufficient to remove the negatively charged particles from an 
atom. 

5. Atoms are made of smaller particles 

6. Calculating the stability of atoms is difficult because of the large number of 
particles involved. 

7. velocity of cathode rays is proportional to the potential difference between the 
cathode and anode. 

8. The material of which the cathode is manufactured is unimportant to the pro- 
cess. 

9. "I can see no escape from the conclusion that they are charges of negative 
electricity carried by particles of matter." 

9.1.2 Fundamental Charge: Robert Millikin 

(notes from Modern Physics: for scientists and engineers by Taylor, Zafiratos and 
Dubson) 



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Thomson could measure the ratio of m/e but not m or e, he measured the ratio 
from the charges iteration with magnetic and electric fields 

ma = -e(E + v x B) (9.16) 

Robert Millikin was able measure the mass of a oil drop for which the mass was 
known and measure the charge on that oil drop. 
Experimental method 

1. Spray a fine mist into the region between 2 charged plates. The drops fall 
reaching terminal speed with the weight of the drop balanced by the viscous 
drag of air. 

2. Switch on an electric field, some drops move down more rapidly, some drops 
move upward. 

The conclusion is some oil drops have acquired positive or negative charges. 

9.1.3 Measuring M by terminal speed 

The terminal velocity may be calculated from Stoke's Law, 

F = 6nr]rv (9.17) 

v = -J— (9-18) 
burr] 

where, r =radius of sphere, i]= viscosity of gas, F = Mg = 4/37rr 3 pg= weight of oil 
drop. Now we write, 

r 2 ^ (9.19) 
4 pg 

Since p, g, r\ are known, a measurement of the velocity gives the radius and the radius 
gives us the mass M. 

After the inclusion of an electric field, 

qE — mg = Q^rjrv2 (9.20) 
V2 = —p, (9.21) 



67T7]r 

V2 qE — mg qE 
v 2 mg mg 



- 1 (9.22) 



Millikin also noticed that occasionally a drop would suddenly move up or down 
indication a change in charge. Millikin theorized that this was due to gaining electrons 
from the ionized air in the chamber. In order to test this he ionized the air by exposure 
to x-rays. In this way he studied the change in charge. 



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Subsequently, he measured that all changes in charge were integer multiples of 
e = 1.6 x 10~ 19 C. This is the basic unit of charge. The charge of an electron. 

Millikin also noticed when there was a near vacuum in the container x-rays changed 
that charge but only to the positive, Leading Millikin to correctly conclude that the 
x-rays are knocking electrons out of the oil drops. 

2 important results, 

1. The charge of one electron is q = — e, this leads to the mass of an electron being 
1/2000 Hydrogen 

2. All charge come in multitudes of e. 



(notes from Modern Physics: for scientists and engineers by Taylor, Zafiratos and 
Dubson) 

The great question after Thomson discovered the electron was, what is the 
structure of the atom? Thomson gave an opinion; Thomson Model: Electrons 
embedded in a sphere of uniform positive charge (blueberries in a muffin). 

To test this charged particles were used to bombard a target of a metallic foil. 
Helium nuclei known as a-particles were used. (1909 Rutherford). 

If a particles strike a thin layer of matter most pass though suffering only small 
deflections. This implies that atoms are mostly empty space. Deflections are caused 
by the electric fields of atoms. Most deflections are small, however a few particles 
were deflected through large angles ~ 90°. Thompson believed that large angles were 
the result of many deflections. Rutherford, however showed that the probability of 
striking one atom was small, and many atoms exceedingly small. This implies that 
the electric field of an atom must be greater than previously believed. 

The electric field may be calculated from, 



and if k is a constant and q cannot change the only way to increase the electric field 
is to reduce the radius r. 

In light of the need for a decreased radius, Rutherford proposed an atom with all 
of the positive or negative charge concentrated into a tiny massive nucleus, with a 
cloud of opposite charge outside of the nucleus, most alpha particles pass far from 
the nucleus those that get close are scattered through large angles. These deflections 
can be explained by coulomb forces. 

Rutherford calculated; 

• The trajectory of an a/p/ia-particle through different angles 



9.2 The Nucleus: Rutherford 



E = 



kq 



(9.23) 



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• Predicted number of deflections through different angles 

• Predicted how this number would vary with foil thickness 

These were verified by Gieger and Marsden 1913 two scientists working with Ruther- 
ford. 

9.2.1 The Rutherford Formula 

[INSERT PICTURE] 

From coulomb's law we find the force on an a-particle of mass m and charge 
q = 2e, from a nucleus of charge Q = Ze can be calculated as, 

F = f (9.24) 

= ^ (9-25) 

where k = 8.99 x 10 9 Nm/C 2 . 

We assume that the collision is perfectly elastic. Similarly the mass of the nucleas 
is so much greater than that of the a-particle, thus we can assume that the magnitude 
of the momentem does not change, allowing us to write the change in the momentum 
of the a-particle as, 

Ap = 2 Pl sin °- (9.26) 
From Newton's second law we write, 

/oo 
Fdt (9.27) 
-oo 

Now we choose x in the direction of the change in momentum, 



Ap 



/oo 
F x dt (9.28) 
-oo 



00 2Ze 2 k 

cos (j)dt (9.29) 

(9.30) 



/ r 2 

J —CO 1 



We can write the initial angular momentum as L = p,ib, and L = mr u = mr this 
allows us to calculate oo = ^ as, 

-r = —o 9.31 
dt mr 1 



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Now we calculate the change in momentum as, 

/°° 2Ze 2 k 
— cos (f)dt, (9.32) 
■oo ^ 

= 2kZe 2 [ ^|^0, (9.33) 



2kZe 2 r** 

bpi 
2kZe 2 



bpi 



r<ps 

/ cos0#, (9.34) 
(sin0/ — sin0j) (9.35) 



Since 0/ = - fc, and 9 + 20/ = 180°, = 90° - 9/2 and sin0/ = cos(0/2). 
Now we write Ap as, 

AkZe 2 m .9. 

Ap = — cos- (9.36 

bpi 2 

= 2p t sm( 9 -) (9.37) 

(9.38) 

This leads to the relationship between the impact parameter b and the scattering 
angle 9. 

where E = p 2 /2m. This equation can be understood as particles with an impact 
parameter V < b will scatter at an angle & > 9. [INCLUDE PICTURE] 

We can now calculate the number of particles scattered between an angle 9 and 
9 + d9. 

If the beam of a-particles has a crossectional area of A. The proportion of atoms 
scattered by 9 or more is irb 2 /A, so that the number of particles scattered is, 

nb 2 

N s = N— (9.40) 
J\ 

Multiply this by the number of target atoms illuminated by the beam, where the foil 
has a thickness of t and contains n nuclei per unit volume. 

n t = nAt (9.41) 

N sc {9' >9) = {^-){nAt) = TiNntb 2 (9.42) 



Now we substitute in b in terms of 9, 

Zke 
VEtan(|) 



iV„ = TiNnt (-—^tjz) (9.43) 



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Now we differentiate with respect to 9, 

dN sc (9 ^9 + d9)= Nntf^-Y °° S ^ d9 (9.44) 

V E / sin (!) 

These particles are scattered across a ring of area, 

dA = (2ns sin 9)(sd9) (9.45) 

then the number of particles scattered between 9 — > 9 + d9 over unit area is, 

Zke 2 \ 2 cos(|) 



nsc[U) dA 2ns 2 sm9d9 ' 

~ 4i2"l~B~J sin 4 (£/2) ( } 

where, sin^ = 2 sin(|) cos(|). Eq (9.47) is known as the Rutherford formula. N 
original number of a-particles, and s 2 is the square of the distance to detector. 

1. n sc {9) oc t thickness 

2. n sc (9) oc (Ze) 2 nuclear charge squared 

3. n sc (9) oc 1/E 2 the incident energy squared 

4. n sc (9) oc l/sin 4 (#/2) 

The fact that the coulomb force always holds gives an upper bound on the size of 
a nucleus. 

Example: upper bound on nucleus size 

assume an a-particle with energy 7.7Mev is fired at gold foil (Z=79), find the upper 
limit on nuclear size. 

The distance of the particle to the nuclear center is always greater than the radius 
of the nucleus, r > R. The minimum value of r will occur for the case of a head on 
collision, at the point of minimum r, the particle has an instanaeous minimum (as the 
particle turns around) in the kinetic energy (K=0). Thus its potential energy here is 
equal to its total energy, 

2Zke 2 

U = = E = 7.7Mev (9.48) 

T 

1 mm 

now the r min is now, 

2kZe 2 



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since R < r for all orbits, 

^ 2kZe 2 

R < -g- (9-50) 

we can write he 2 = lAAMeVfm, this allows us to calculate, R < 30 fm 

We can see that as the energy increases, the rutherford formula breaks down as 
the minimum value of r approaches the actual value of R. The energy at which the 
rutherford fromula first breaks down is 

E ~ ^ (9-51) 



Homework 23 

(3.49 in Taylor) (a) If the Rutherford formula is found to be correct at all angles when 
15-MeV alpha particles are fired at silver foil (Z=47), what can you say about the 
radius of the silver nucleus? (b) Aluminum has atomic number Z=13 and a nuclear 
radius Rai ~ 4 /to. If one were to bombard aluminum foil with alpha particles and 
slowly increase their energy, at about what energy would you expect the Rutherford 
formula to break down? [You can make the estimate a bit more realistic by taking R 
to be Rai + Rue where Rn e is the alpha particles radius (about 2 fm). 



9.3 The Bohr Model 

By 1870 the atomic spectra from emission and absorption was comonly used to iden- 
tify elements. There was however no satisfactory explaination for the spectra from 
classical physics. 

By 1885 the four visible lines of the hydrogen molecule had been accurately mea- 
sured, and swiss schoolteacher Johann Balmer had shown that these wavelengths fit 
the heuristic formula, 

\-*&-h) ^ 

where n = 3, 4, 5, 6 for the visible lines, and R is a constant of value R = O.OIIOto 1 . 
This was generalized to, 




(9.53) 



This equation is known as the Rydberg formula and describes all wavelengths in the 
hydrogen spectra. 

A satisfacory model would have to wait until 1913 when Niels Bohr published "On 
the Constitution of Atoms and Molecules". Bohr postulated that electrons are bound 
in stable energy levels, and energy is only emitted when an electron passes between 
levels. From this Bohr was able to predict the Rydberg formula. 



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Bohr's model of the atom was a refinement of Rutherford's model. This model 
is planetary, with the nucleus functioning as the sun and electrons as the planets. 
Under classical electro-magenetic theory if a particle is accelerating, then it is emit- 
ting electromagnetic waves. When a particle is radiating it is losing energy and the 
orbit will decay at a rate of 10 _11 s. Thus under classical electro-magenetic theory 
Rutherford's model is unstable. 

Bohr proposed as a refinement, only a certain discrete set of orbits are possible. 
This leads to two implications, 

• the energies of such orbits are discrete, 

• the electron only radiates when passing between discrete orbits. 



9.3.1 The Bohr radius 

Bohr assumed the orbits where close to classical orbits, then from the coulomb force 
we find, 

hp 2 

m- = ^ (9.55) 

classically there is no restriction on v, or r which may range between and oo. we 
can write the potential and kinetic energy as, 

K = \nv 2 (9.56) 
hp 2 

U = -— (9.57) 

r 

At infinity U — this leads to the kinetic energy and total energy being, 

K = -hj (9.58) 
1 1 ke 2 

E = K + U = -U = -- (9.59) 

_ Zi T' 

This result is commonly known as the viral theorem. 

Bohr theorized that the orbits are discrete, meaning that like energies the angular 
momentum is quantized. 

Bohr then postulated that the angular momentum, L is quantized and propor- 
tional to plank's constant. 

l = A 2 A, 3 A )4 A,... (9 . 60 ) 

2tt' 2tt' 2tt' 2tt' v ; 

L = nh (9.61) 



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where H = h/2n = 1.054 x 10~ 34 Js. Now can write the angular momentum as, 

mvr = nh (9.62) 
nh 

v = — . 9.63 

mr 

Calculate the radius of the orbit, from the conservation of energy, 

! ' (9.64) 

(9.65) 
(9.66) 

= n 2 a . (9.67) 



K = 


-2 U ' 


1 2 


Ike 2 


-mv = 
2 


27' 




ke 2 


m( — y = 




mr 


r 


r = 


2 ^ 

ke z i 



where a = = 0.0529nm is the Bohr radius or the radius of the hydrogen atom 
in its ground state. This is not exact because electron positions are merely probabilty 
densities, but it is very close to the average position of the electron. 



9.3.2 The energy of the Bohr atom 

Knowing the possible radii we can calculate the possible energies, 

E = -|- (9.68) 
ke 2 1 

; , where n= 1,2,3... (9.69) 



2ao n 2 



Thus the possible energies are quantized, and the energy emitted when dropping from 
one energy level to another is, 

E 1 = E n -E' n (9.70) 

" 2 (^) 



2a 







n' 2 n 2 



= hcR[---) (9.72) 
which is the result given by Rydberg. 



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Now we calculate R from the values here, 

hp 2 

R = (9.73) 
1a hc v ; 

= < 9 - 74 > 

lAAeVnm 

(2) (0.0529nm) (1240e Vram) ^ ' ^ 

= 0.0110m" 1 (9.76) 

note a = 1/137 is a constant known as the fine structure constant. One finds the 
value found from Bohr's theory for R is equivalent to the experimental value. 
We define the Rydberg energy as, 



(9.78) 



E R = hcR (9.77) 
2a 

= ^ (9.79) 

= 13.6eV (9.80) 



when n — 1, now we can write the energies of each orbit as, 



K = ~ (9.81) 



n 2 

The lowest possible energy corresponds to n — 1 this is known as the ground state, 
states with higher energies are known as excited states. 

• The ground state energy corresponds to the energy required to remove an elec- 
tron entirely from an atom (in this case from hydrogen) 

• excellent agreement with experiment 

• r = ao = 0.0529nm average radius of electron orbit 

• Generalized Bohr's theory gives a good order of magnitude estimate of other 
atomic radii. 



[INSERT PICTURE] 



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9.3.3 Generalizing the Bohr Model 

we now have nuclear charges of Ze thus we write the force as, 

p= zkl_ 

r 

This allows us to calculate the radius as, 

h 2 



n 2 | (9.84) 



(9.82) 



And we find that the energy of the nth orbit is given as, 

E n = -Z*f\ = -Z^ (9 . 85) 
2a n 2 n 2 

For a He + ion the photons emitted are given by, 

^M^-h) (9 ' 86) 

This has been observed in solar spectra. 

We should note that because there real center of mass of the atom is not coincident 
with the center of the nucleus, we can correct the Rydberg energy by, 

_ M^e 2 ) 2 



where ji is the reduced mass, 



777 

fj, = - — (9.88) 

l+m/m n 



m is the mass of an electron and m n is the mass of a nucleus. 



Homework 24 

(5.11 in Taylor) Find the range of wavelengths in the Balmer series of hydrogen. Does 
the Balmer series lie completely in the visible region of the spectrum? If not, what 
other regions does it include? 

Homework 25 

(5.13 in Taylor) The negative muon is a subatomic particle with the same charge as 
the electron but a mass that is about 207 times greater: m M ps 207m e . A muon can be 
captured by a proton to form a "muonic hydrogen atom" , with energy and radius given 
by the Bohr model, except that m e must be replaced by m M . (a) What are the radius 
and energy of the first Bohr orbit in muonic hydrogen? (b) [not assigned] What is 
the wavelength of the Lyman a line in muon hydrogen? What sort of electromagnetic 
radiation is this? (Visible? IR? etc.) Treat the proton as fixed. 



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9.4 The Neutron 



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Quantum mechanics III: Elements 
of Quantum Theory 



10.1 Compton Scattering 

When a beam of light is fired at a system of charges, some of the beam is scattered. 
The classical theory says the oscillation electric field of the incident light causes the 
charges to oscillate, and the oscillating charges then radiate secondary radiation in 
various directions. The frequency / of the scattered waves must be the same as 
that of the oscillating charges, which must be the same as the incident frequency / . 
Classically / = f . 

Starting in 1912 reports of High frequency x-rays being scattered of electrons, 
where the scattered frequency was less than the incident frequency /< Jo- 
in 1923 Arthur Compton argued that if light is quantized, one should expect 
f < fo- Since photons carry energy they should also carry momentum. From the 
Pythagorean energy in special relativity, 

E 2 = (pc) 2 + (mc 2 ) 2 , (10.1) 

we calculated the energy of a massless object as, 

E=pc. (10.2) 

From the photoelectric effect E = hu allowing us the write the momentum as, 

E his h 

P= -c = - = X (1 °- 3) 

Thus in a collision with a stationary electron there is a transfer of energy and mo- 
mentum, causing a reduction in the energy of a photon. 

Naturally, the electrons are not purely stationary, the energies of outer shell elec- 
trons are on the order of leV. While, the energies of an x-ray photon correspond to 



81 



Chapter 10 Introduction to Modern Physics: Physics 311 

wavelengths between O.OOlnm < A < lnm, or energies between, 1.2MeV < E < 
1.240Met> . The energy of the electron is small enough to neglect in terms of calcu- 
lation. [INSERT PICTURE] We will calculate the results using an elastic collision 
where, energy and momentum are conserved. 

E e + Ei = mc 2 + E (10.4) 

Pe+Pl = Po (10-5) 

E e = mc 2 + E -E 1 (10.6) 

Where E e ,p e is the energy and momentum of the electron after the collision, £"o,po is 
the energy and momentum of the photon before the collision and Ei, p\ is the momen- 
tum and energy of the photon after the collision. We treat the electron relativistically 
by writing E = p Q c and E\ = p\c. Now we write the conservation of energy as, 

E e = mc 2 + E -E l (10.7) 
\/ p 2 c 2 + [mc 2 ) 2 = mc 2 + poc — pc (10.8) 
\/p 2 + (mc) 2 = mc + po—p (10.9) 

(10.10) 

Now we wish to replace p e with a function of the photon momentum, by using the 
conservation of momentum equation. 

Pe = P0-P (10.11) 
P\ = Pe-Pe = (P0-P) ■ (P0-P) (10.12) 

= p 2 +p 2 -2p -p (10.13) 
= Pi + P 2 ~ ZpoP cos 6 (10.14) 

(10.15) 

Now, 

\J p 2 + (mc) 2 = mc + po—p (10.16) 
pi + (mc) 2 = (mc + po—p) 2 (10.17) 
pi + p 2 — 2poP cos 9 + (mc) 2 = (mc) 2 + 2p mc — 2pmc — 2p p + pi + pflO. 18) 
mc(p — p) = p p(l — cos9) (10.19) 

— - - = — (1 - cos 6) (10.20) 
Po p mc 

since p = hu/c, we write, 

' ' /; (l-cos0) (10.21) 



v\ u mc 2 



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or in terms of the wavelength we write, 

AA = Ai-A = — (l-cos0) (10.22) 
mc 

We observe that the shift depends on the angle. We can replace the term h/mc by 
the Compton wavelength \ c = h/mc = 0.00243nm. 

A 1 -A = A c (l-cos^) (10.23) 

• ^max occurs at 9 — 180° 

Delta\ max = 2A C = 4.86 x 10 _3 nm (10.24) 

• Since this is the max charge, A's that are several thousand times larger are hard 
to measure the Compton shift. 

• Compton effect negligible for UV, visible, IR, only noticeable for x-rays, 7-rays 

• Einstein and Compton 

- light is a particle 

— dual nature = wave-particle duality 



10.1.1 Example: Compton Scattering 

Calculate the percentage change in A observed for Compton scattering of 20keV pho- 
tons at 9 = 60°. 

A1-A0 = A c (l -cos 9) = 1.22pm (10.25) 
he 

A = — = 0.062m = 62pm (10.26) 
E 

^ x 100% = 1.97% (10.27) 
Ai 

Now calculate the maximum precent change for the A = 0.0711nm x-rays used by 
Compton in his scattering experiments off of graphite electrons. The maximum cor- 
responds to the maximum change in kinetic energy of the electron, or the greatest 
change in wavelength where 9 = 180°. 

AA = A c (l-cos^) = 0.00486nm (10.28) 

AA M 0.00486nm „ mM 

x 100% = — — = 6.8% « 7% 10.29 

Ai 0.0711nm v ' 

Calculate the maximum percent change for a visible light at A = 400nm, 

AA = A c (l-cos^) = 0.00486nm (10.30) 

AA M 0.00486nm w 

— x 100% = w 1 x 10" 3 % 10.31 

Ai 400nm 



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Homework 26 

(Tipler) Compton used photons of A = 0.0711nm (a) What is the energy of these 
photons? (b) What the A of the photons scattered at 9 = 180°? (c) What is the 
energy of the scattered photons? 



Homework 27 

(4.31 in Taylor) If the maximum kinetic energy given to the electronic a Compton 
scattering experiment is lOkeV, what is the wavelength of the incident x-rays. 



10.2 de Broglie 

Compton showed that a photon displays both the properties of a wave E = hf and 
the properties of a particle p = h/X 

de Broglie postulated that material particles have a similar particle-wave duality, 
and as such they should have a wavelength defined by 

A = -, (10.32) 
P 

10.2.1 Example: de Broglie wavelength 

Find the de Broglie wavelength of a particle of mass lCT 6 g moving at a speed of 
v = 10~ 6 m/s. 

A = - = — (10.33) 

p mv 

6.63 x lO" 34 Js iq 

= 7 a — rr = 6.63 x 10~ 19 m 10.34 

(10- 9 %)(10- 6 m/s) v ; 

Since the wavelength is much smaller han any possible obstacles diffraction or inter- 
ference of such waves cannot be observed. 

Two important experiments in 1927 established the existence of the wave proper- 
ties of electrons 

1. Davisson, Germer Electron scattering from a nickel target. The intensity of 
scattered electrons varied with angle, showing the results of diffraction. 

2. CP. Thompson (Son of J.J.) Observed electron diffraction in the transmission 
through metal foils 

The calculation of the wavelength from observed diffraction is the same as the de 
Broglie wavelength. 

Diffraction of electrons leads to the electron microscopes. 



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Homework 28 

(Tipler) A proton is moving at v — 0.003c, find its de Broglie wavelength. 
10.2.2 Quantization of angular momentum 

de Broglie argued that if electrons were a wave then the angular momentum would 
be quantized in multiples of h. A wave can then fit into a circular path, if the 
circumference can accommodate and integral number of As. 

n\ n = 1, 2, 3, . . . 



27IT = 


nX 


h 




P 






h 


2nr = 


n— 




P 




nh 


rp = 


2^ 


L = 


nh 



(10. 


.35) 


(10. 


,36) 


(10. 


.37) 


(10. 


.38) 


(10. 


.39) 


(10. 


.40) 



Which is the same result as Bohr assumptions in his model of the hydrogen atom. 

10.3 Wave-particle duality 

Light and matter both exhibit qualities of both waves and particles. Which is true? 

de Broglie believed the relationship was such that waves can behave like particles 
and particles like waves. 

10.3.1 Young's Double slit Experiment 

Imagine two slits. 

Particles striking 2 slits. 

1. come in lumps 

2. probability of hitting is the sum of the probability from each slit, 
waves striking 2 slits 

1. continuous 

2. measure intensity 

3. Intensity from 2 slits not the sum of the intensity from each slit 



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4. shows interference, 
electrons striking 2 slits 

1. come in lumps 

2. measure probability of hitting 

3. probability from 2 slits not the sum of the probability from each slit 

4. shows interference. 

We observe this interference pattern even for 1 electron, meaning it interferes with 
itself. The experiments show p = h/X. 

A quantum particle can be described as a 'wave' that interferes with itself. In 
fact, we cannot predict where the electron strikes. 

This is a fundamental shift to a probabilistic dynamics rather than a deterministic 
one. 

We can determine a distribution of probability, which tells us the chance that a 
quantum particle hits a particular spot. In the Schroedinger description of quantum 
mechanics this probability amplitude is called the wave function ip. 



The wave function for a matter wave is designated by the greek letter psi ip. The 
motion of a single electron is described by the wave function ip, which is a solution 
to the Schrodinger equation and contains imaginary numbers. The wave functions 
are not necessarily real. The probability then of finding an electron in some region of 
space is | tp | 2 . We call this the probability density, 



1926 Edwin Schroedinger (Austrian) Presented a general equation that describes 
the deBroglie matter waves, 



K, U, and E are operators that relate to the kinetic energy potential and total energy, 
using calculus we can calculate ip. 

For a hydrogen atom in a ground state n = 1 the maximum of | ip | 2 corresponds 
to the Bohr radius. 



10.4 Probability Waves 




(10.41) 



(K + U)i> = E^/j 



(10.42) 



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10.4.1 Example: Probability density 

A classical particle moves back and forth with constant speed between 2 impeneratble 
walls at x = and x = 8cm. (a) what the probability density P(x)l (b) What is 
P(x = 2cm)? (c) What is P(x = 3 ->■ 3.4cm)? 
(a) 

P(x) = P < x < 8cm (10.43) 
P dx = 1 (10.44) 



o 

The probability of finding the particle in the interval dx at point x\ or at point x 2 
is the sum of the separate probabilities P(xi)dx + P(x 2 )dx. Since the particle must 
certainly be somewhere, the sum of the probabilities over all possible values must 
equal 1. 

P(x)dx = / P dx = 1 (10.45) 



J o 

^=sL < io - 46 > 

(b) Since dx = The probability of finding a particle at the point x = 2cm is 0. 

(c) Since the probability density is constant the probability of a particle being in 
the range Ax is P Ax. The probability of the particle being in the region 3.0cm < 
x < 3.4cm is thus, 

P Ax = ( — )0.4cm = 0.05 (10.47) 

v 8cm ; v ; 



10.5 Electron wave packets 

A Harmonic wave on a spring is represented by the wave function 

y(x, t) = A sm(kx - cut) (10.48) 

Where k = 2tt/X is known as the wave number, and the angular frequency is oo = 2irf. 
The velocity of the wave then is given by, 

.-/*-£><!>- 

u, k have no limit in space-time. 

One electron that is localized in space must be explained by a wave packet. In 
order to generate a wave packet that is localized in space we need a group of harmonic 
waves containing a continuous distribution of k, u. 



(10.49) 



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Consider a grouping of 2 waves, 

if){x, t) = A sin(A; 1 a; — ujit) + A sm(k 2 x — u 2 t) (10.50) 

using, 

sin^ + sinfl 2 = 2cos(^^ 2 ) sin(^^ 2 ) (10.51) 
We can write the wave as, 

11 11 

ip(x, t) = 2A cos(-Akc - -Auxt) sm(-k ave x - -u ave xt) (10.52) 

This wave describes the superposition of two waves with velocitie v — ^ and v = 

If we place x\ at the position where both waves are zero and x 2 at the next such point, 

then we know that, 

^Akx 1 - ^Akx 2 = vr (10.53) 

which can be rewritten as, 

AkAx = 2tt (10.54) 

In General, 

ip(x, t) = ^ Ai sva{kiX — u>it) (10.55) 

i 

• To solve AiS requires Fourier analysis 

• Requires A continuous distribution of waves 

• Replace with A(k)dk 

The group velocity of the wave packet becomes 

= | (10.56) 
We need to find u>(k), Starting with the Energy and momentum, 

E = hf = h^ = Huj (10.57) 

p = - x = ^ = hk (10.58) 
Now since we can write the kinetic energy as, 

2 

E = — (10.59) 

2m v ; 

to = ^ (10.60) 
"0 = £ (10-M) 



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Now we find the group velocity as, 

dio d /hk 2 \ hk p (10 62) 

Vg ~ ~dk ~ dk\2m) ~ 2m ~ m ~ V ^ ' ' 

where v is the velocity of the electron. 

The phase velocity of the individual waves inside the wave packet is equal to, 

v v = — = — = — = — = - 10.63 
k hk p 2m 2 K J 

[INSERT FOURIER ANALYSIS PICTURES] The standard deviation of the Fourier 
transforms of these Gaussians are, 

o x a k = X - (10.64) 

10.5.1 Heisenberg Uncertainty 

• consider an electron with wave packet ip(x,t) then the most probable position 
is given by the maximum of | <ft(x, t) | 2 . 

• over a series of measurements over electrons with identical wave functions, we 
will measure, | (fi(x,t) | 2 . 

• If the distribution in position is narrow, the distribution in k must be wide, 

• A narrow wave packet corresponds to small uncertainty in position and a large 
uncertainty in momentum 

In general the ranges Ax and Ak are related by, 

AkAx ~ 1 (10.65) 

Similarly, a wave packet that is localized in time must contain a wide range of fre- 
quencies, 

AujAt ~ 1 (10.66) 
If we multiply these results by h then, 

ApAx rsj h (10.67) 
AEAp ~ h (10.68) 

If the standard deviations are defined to be Ax and Ap, we know from the Fourier 
transform that the minimum value of their product is l/2h. Now, 

ApAx > - (10.69) 

AEAp > ^ (10.70) 



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10.5.2 Schroedinger's Equation 

How did Schrodinger arrive at his wave equation? Normally in a wave equation a 
second order time derivative is paired with a second order spacial derivative, 

8*=*W (1 °- 71) 

Schrodinger's equation involves only a first-order derivative in time. This was neces- 
sary in order to display de Broglie waves in the non-relativistic limit. 

* + U{x)* = ih 9 4 (10.72) 



2m dx 2 dt 

Under certain circumstances we can write the wave function as the product of a time 
dependent function, and a time independent function, 

ij>(x, t) = ip(x) exp[-iwt] (10.73) 

Now we can see if a function of this form is placed into the Schrodinger equation, 

h 2 d 2 tb(r) 

e- iw \-^^p^ + U(x)4>(x)) = {-iu){ih)e- iwt (10.75) 

h 2 d 2 ib(x) , , . . , , . 

- — -£y- + U(x)il>(x) = E^j(x) (10.76) 

leads to a time independent Schrodinger equation, after the elimination of the time 
variable. 

There are several important conditions to the wave equation, 

• ifj(x) must be continuous in x. 

• U(x) not infinite 

• ^ is continuous 

ax 

• We also impose a normalization condition, 

/oo 
I ip(x) | 2 dx = 1 (10.77) 
■infty 



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10.5.3 Particle in a one-dimensional box 

A free quantum particle satisfies the Schrodinger equation, 



h 2 d 2 ^ dip , n7fi , 
ih— (10.78) 



Let us try the solution, 
Then, 



2m dx 2 dt 
ip(x,t) = <f>{x)e- iE/h (10.79) 



e^(-^g) = m-^)e-^(x) (10.80) 
h 2 <9 2 



2m dx 2 



E<j>(x) (10.81) 



ft 2 # 2 , 

-E(f)(x) = (10.82) 



2m dx 2 

}2 



d 2 <p 



k 2 (f)(x) = (10.83) 



dx 2 

The solution of this equation is clearly, 

(f)(x) = Aexp[±ikx] (10.84) 

[check it, if it satisfies the equation the uniqueness theorem says its true] 

Now let us consider if we place an impenetrable box of dimension a between, x = 

and x = a, this implies the boundary conditions of, 0(0) = 0(a) = 0. 
The most general solution is now, 

(f)(x) = Aexp[ikx] + B exp[— ikx] (10.85) 

where k = ^j2mE/h 2 and A, and B are constants, Applying the boundary condition 
0(0) = 0, 

(f)(0) =A + B = (10.86) 
There from A=-B, and we can write the solution as, 

<f>(x) = C sin kx (10.87) 
Where C = 2%A. Now we impose the boundary condition 0(a) = 

sinA;a = (10.88) 
Therefore, ka = nir, This means that the wave number is quantized, 

k n = — (10.89) 

(X 



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and so is the energy, 



^ h 2 k 2 n n 2 7i 2 h 2 



with n — 1, 2, 3 • • • 

This quantization is a result of the boundary conditions imposed on the Schrodinger 
equation. We now write the wave function as, 

Tl 7TT 

n = Csin— (10.91) 

(X 

We normalize this, 

pa 

<p n | 2 dx = 1 (10.92) 



f 

Jo 



and we find that C = a/2/<2, 



/ 2 . nnx n _ x 

n = A /-sin (10.93) 



a a 

The complete solution is given as, 

^ n (x, t) = y - sin exp[— 2-^— J (10.94) 

And the probability of finding a particle at position x at in a stationary state ip n (x, t) 
is, 

P n {x,t) =\ M^,t) \ 2 = - sin 2 — (10.95) 
Since the Schrodinger equation is linear, a superposition of states is also a solution, 

ijj(x,t) = y|foMM) +VV'0M)1 (10.96) 
We calculate the the probability P(x,t) for such a wave, 

P„0M) =| ^„(x,t) | 2 = -[sin 2 ^^ + sin 2 ^^ (10.97) 



a 



a a 



+2 sin sinn racos (10.98) 

a a 

There is now oscillatory behaviour between this two states, the probability changes 
with time, and with a frequency of, 

v = {En ' ~ En " ] (10.99) 

which corresponds to the Bohr frequency. 
Some additional points 



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1. the number n is know as a quantum number as it specifies which quantum state 
the system is in. 

2. There is a certain minimum energy the system must have, E\ = 

3. The distance between energy levels increases with decreasing a, as a gets larger 
we approach the classical limit of continuous energy. 

10.5.4 Tunneling 

• Some regions are forbidden by energy considerations 

• U > E T ot particle would have to have a negative K 

• Limited by potential barrier 

• There is a small but finite probability of the particle penetrating the barrier: 
tunneling. 

10.6 Spin 

We all know that the world is not 1-dimensional, so we would like to write out (not 
solve) the 3-dimensional Schrodinger equation. 

If we combine this to a box of dimension Lx Lx L, then we can write the energies 

as, 

E = ^Ak\ + k\ + kj) (10.101) 



h 2 7T 2 

2mL 2 



ni + n 2 2 + n 2 3 ) (10.102) 



We can see there is a quantum number for each dimension. 

Similarly we imagine that the confining potential of a [hydrogen] atom is not 
a cube, but rather a a spherically symmetric potential, thus we need to write the 
Schrodinger equation in spherical coordinates. The transformation need not concern 
us, 

h 2 \ I d { ?dib \ 1 d ( . n dib\ 1 d 2 ibi TT . „ , 

-*d r i r ) + ^rem( 3me m) + *r B m + = (10103) 



2m 



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Solutions for the spherical Schrodinger equation may be written as the product of 
functions for separate variables, 

il>(r,e,il>) = R{r)f{9)g{4>) (10.104) 

often however the two angular coordinates are coupled such that the solution becomes, 

Vw(r, 9, V>) = R nl (r)Y lm (0, <f>) (10.105) 

Where R n i{r) are related to Laguerre polynomials, and Yi m (0,(j>) are spherical har- 
monics, 

V ; LVnan/ 2n n + l ! 3 J Vna n / n+1 Vna n / ; 



>WM = [ ^ / + m , ] (-l)"'e'"'*Pr(co S e) (10.107) 
™ - ^W^.^-^^-l)' (10.108) 

Now I included here a lot of unimportant details, but what we notice is that there are 
still 3 quantum numbers, however now nlm are interdependent. The possible values 
of these quantum numbers are, 

n = 1,2,3- •• (10.109) 
I = 0,1, 2,--- ,n- 1 (10.110) 
m = -/,-/ + l,-/ + 2,--- ,+/ (10.111) 

n is known as the principle quantum number, 1 the orbital quantum number, m the 
magnetic quantum number. 

When the spectral lines of hydrogen are viewed under high resolution, it is found 
to consist of two closely spaced lines. This splitting of a line is known as fine structure. 

In 1925 W. Pauli suggested the electron has another quantum number that can 
take 2 values. 

S. Goudsmit and G. Uhlenbeck (1925) suggest this quantum number is the z— component 
of the angular momentum of the electron with values, m s — 2s + 1, s — ±1/2. 



10.7 The Copenhagen Interpretation 



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Structure 



When the S.E. was solved for hydrogen the results predicted the energy levels of the 
Bohr theory. 

• Bohr model predicted one quantum number, n - principle quantum number 

• SE supplied 2 other quantum numbers, 

— 1 - the orbital quantum number 

— m - the magnetic quantum number 

• m shows up only when atoms are placed into a magnetic field. For a given, /, 
the level is split when placed into a magnetic field, into mi levels. 

• under high-resolution optical spectrometry each emission line is actually 2 lines. 

• this splitting is known as fine structure. This spitting is characterized by the 
electron spin quantum number - m s . 

• Spin orientation is with respect to the atom's internal magnetic field, produced 
by orbit of an electron. 



Quantum number 



number of allowed values 



Quantum numbers / 



mi 
m s 



n 



1,2,3,... N/A 

0, 1, 2, . . . , (n — 1) n for each n 

0,±1,±2,±3,...,±Z 21 + 1 for each/ 

±1/2 2 



Other particles have spin. 



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11.1 Multi-electron Atoms 

The Schroedinger Equation cannot be solved exactly for multi-electron atoms. We 
can however approximate multi-electron atoms with a hydrogen atom, the energies 
will depend on n, I. 

• Electrons that share the same quantum number n make up a shell 

• The sub-shell is defined by the quantum number /. It is common to use letter 
instead of numbers for the / quantum numbers, 

s p d f g • • • 
1 2 3 4 ... 

an electron that is 2p, n = 2 and I = 1, 3d, n = 3, I = 2. 

The numerical sequence of the filling of the shells depends on the energy of the 
shells, [INSERT PICTURE] we notice that 4s is below 3d. 
How do electrons fill shells? 

11.1.1 Pauli Exclusion Principle 

Pauli Exclusion Principle: No two electrons in an atom can have the same set of 
quantum numbers. 

Thus for n = 1 I = (Is) there is only one mi = therefor there must be another 
unique quantum number m s = ±1/2. There are only two electrons in this shell. 

11.2 Periodic Table 

The periodic table is a tool that is organized by quantum numbers. The vertical 
column are grouped to group elements of similar chemical properties. The electrical 
properties depend on the number of elections in the outer shell of the ground state. 
The rows (or periods) list the elements in increasing atomic number (number of 
protons) 

We notice that the first period has 2 elements, periods 2-3, have 8 elements and 
period 4-5 have 18 elements. 

Let us now think about quantum numbers, and how many electrons can have each 
quantum number. 

• n = 0, I = 0, mi = 0, m s = ±1/2 so there are 2 electrons with n = 

• n — 1, I — 0, mi — 0, m s — ±1/2 so there are 2 electrons with n — 1, I — 0; 
I — 1 m — — 1, 0, 1 so there are 6 electrons with n—\,l — \ 

• n — 2 I — 0, 1, 2, there are 2 electrons with n — 2, 1 — 0, 6 with n — 2, 1 — 1 and 
10 with n = 2,1 = 2. 



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We can see that from this there should be 2 elements in the first period, 6 in the 
second and 18 in the third. Why the discrepancy, this is because the 4s level is lower 
in energy than the 3d level the 4s fills first and so the 3d elements are placed into the 
4th period. 

We can use the periodic table to express the quantum numbers of the ground 
state of an element. For example, Nickel, Ni, ls 2 2s 2 2p 6 3s 2 3p 6 4s 2 3<i 8 . The superscript 
indicated the number of electrons in the shell. 



In quantum mechanics on of the most important questions deals with the processes of 
measurement. Classically, physics is based upon determinism if we know the position 
and velocity of a particle at a particular time its future behavior can be predicted. 
Quantum theory is based on probability we stated earlier that | tp | 2 give the probable 
position of the particle. Because of the probabilistic formulation there is a limit to 
the accuracy of any measurements. 

In 1927 Werner Heisenberg, wrote a second approach to quantum mechanics that 
was later shown to complement the Schroedinger approach. 

Heisenberg Uncertainty Principle: It is impossible to simultaneously know a 
particles exact position and momentum. 

if we write the momentum as p = hk. 

• consider an electron with wave packet ip(x,t) then the most probable position 
is given by the maximum of | <p(x,t) | 2 . 

• over a series of measurements over electrons with identical wave functions, we 
will measure, | (p(x,t) | 2 . 

• If the distribution in position is narrow, the distribution k must be wide, 

• A narrow wave packet corresponds to small uncertainty in position and a large 
uncertainty in momentum 

In general the ranges Ax and Ak are related by, 



Similarly, a wave packet that is localized in time must contain a wide range of fre- 
quencies, 



11.3 Heisenberg Uncertainty 



AkAx ~ 1 



(11.1) 



AuAt ~ 1 



(11.2) 



If we multiply these results by h then, 



ApAx 
AEAp 



h 



h 




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If the standard deviations are defined to be Ax and Ap, we know from the Fourier 
transform that the minimum value of their product is 1/2H. Now, 

ApAx > ^ (11.5) 

AEAp > - (11.6) 
2 

A photon used to observer a particle changes its momentum. 



11.4 Relativistic Quantum Mechanics 

In 1928 P Dirac combined special relativity and quantum mechanics. One result was 
the prediction that for every charged particle the is an identical oppositely charged 
particle. 

• positron charge +e mass= m e 

• electron charge — e mass= m e 

Positrons are created by pair production, electrons and positrons created together 
from the conservation of charge, x-ray passes nucleus creating +e and -e. 
we can calculate the minimum energy needed from. 

E min = hf = 2m e c 2 = 1.022MeV (11.7) 

Positrons captured when passing through matter join with an electron to create 
positronium this decays (« 10~ 10 s) into 2 photons. 



11.5 Nuclear Structure 

After the discovery of the electron the next question remaining was what is the 
structure of the atom. The common model now is based on the Rutherford-Bohr 
model, which is based upon the solar system. Most of the mass is contained in a small 
centrally located nucleus, with electrons orbiting around the nucleus like planets. 

• Concept of a nucleus is based on experiments of the scattering of a— particles. 

• An alpha particle is a helium nucleus, these are naturally produced by some 
radio-active decay. 

• a— particles fired at a thin sheet of gold foil. Since the mass of the a— particle 
is much greater than that of an electron, they will not scatter off of electrons, 

m a > 7000m e (11.8) 



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A simple estimate of size of a nucleus is given by its closest approach. We write 
the electrical potential as, 

L t = fc gig2 = k(2e)(Ze) 

T'rnin Train 

The charge of a nucleus is +Ze where Z is the number of protons. 
Set this equal to the kinetic energy, 

1 2 k(2e)(Ze) 

- mv 2 = — L (11.10) 



2kZe 2 
mv 2 



(11.11) 



Rutherford found the radius of gold to be on the order of 10 m 

11.5.1 Nuclear Force 

The forces between nucleons; 

• Gravitational 

• Electric force (between protons) Since the E. Force is I than the gravitational 
force we need another attractive force to keep the nucleus stable. 

• Nuclear force: strongly attractive l than grav. and e. force, very short range, 
distance (~ 10~ 15 m). 

• Nuclear Force not related to charge. 

11.6 Nuclear Notation 

• The number of protons determines the species of the atom. This is usually 
expressed by a chemical symbol, 



Z+N 

z 



X N (11.12) 

Some times we only express the atomic mass (Z+N) since the chemical symbol 
gives Z. then we might write carbon as, 



12 C (11.13) 

Some atoms may contain a different numbers of neutrons, and the same number 
of protons these are known as isotopes. Isotopes of carbon are, 

12 C, 13 C, u C, n C, 15 C, 16 C (11.14) 

Some Isotopes are more stable than others. The common isotope is the most 
stable. In carbon only 12 C, 13 C are stable. 



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• Isotopes of hydrogen; 1 H, 2 H, 3 H. 

• 2 H is known as deuterium (chemical symbol D) and forms heavy water D2O. 

11.7 The Laser 

The laser is a result of theoretical exploration. LASER: Light Amplification by Stim- 
ulated Emission of Radiation Usually an electron makes a transition to a lower energy 
level almost immediately 10~ 8 s. Some lifetimes can be appreciably longer. A state 
with a long life time is called "metastable" Materials with long metastable states 
"glow-in-the-dark" after the excitation is removed, this is called "Phosporescence" . 

11.7.1 Absorption and Spontaneous Emission 

• Photon absorbed and emitted almost simultaneously 

• If the higher state is metastable 3 possibilities, 

1. Absorption 

2. Emission 

3. Stimulated Emission 

• Stimulated photon identical to the first photon 

• The medium must be prepared, such that more atoms are in the metastable 
state then not. (poulation inversion) 

• Mirrors places at the ends of the medium to enhance spontaneous emission 

• Produces monochromatic/ coherent light 

• because of coherence - very little dispersion. 

11.7.2 Holography 

[INSERT IMAGE] 



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12.1 Radioactivity 

Nuclei that are unstable break down 

• Spontaneously discintegrate or decay 

• they are termed radioactive 

• Decay at a fixed-rate (do not discintegrate at the same time) 

• Discovered by Henri Becquerel 1896 - Uranium (U) 

• 1898 Pierre and Marie Curie, discovered Radium, Polonium 
3 Types of Radiation: 

• a (+2e) Helium nucleus 2p+2n 

• j3 (— e) electron 

• 7 high energy quanta of electromagnetic energy (photon) 

Because of the different charges the radiation can be distiquished by passing it through 
a magnetic field. 

12.1.1 Alpha Decay 

When an a-particle is ejected the mass number is reduced by 4 (AA = —4 and 





(12.1) 



• conservation of charge 



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12.1.2 Example 

U-238 decays by an alpha particle what is the daughter nucleus? A=238, Z=92 after 
the decay Z=90, A=234. The daughter nucleus is Thorium-234 

IfU ^ IfTh+iHe (12.2) 
IfU ™Th + a (12.3) 



The kinetic energy of the alpha particle is typically a few MeV. 2 38 



214 Po 7.7MeV 
U => 4.14M e y 




Alpha Particle 
(Helium Nucleus) 



Since the alpha particles have less energy than the barrier, classically they cannot 
pass. Quantum mechanically we see that the alpha in U-238 commonly crosses the 
barrier. The answer is due to tunneling. Tunneling is the name we give to the fact 
that there is a non-zero probability of an a— particle initially on the outside being 
found inside of the barrier. 

12.1.3 Beta Decay 
• P- 

— An electron is emitted 

— Electron is created during the decay 

— the new negative electron is emitted — > f3— 



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Beta Particle Radiation 



Daughter 
Nucleus 
Calcium-40 




Potassium40 



Beta Particle 



— A neutron decays to a proton and an electron 

— occurs when too many neutrons as compared to protons 

— neutrino also emitted 

l 4 C -> \ A N+\e (12.4) 
In -> \p+\e (12.5) 

• (3+ positron decay 

— too many protons relative to neutrons 

— neutrino also emitted 

— the new positive electron is emitted — > /3+ 

— positron is emitted and neutron created. 

f0 7 fN,+\e (12.6) 

\ V In+U (12.7) 

• Electron Capture 

— Absorbtion of orbital electron 



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result similar to positron decay 
proton changes into a neutron. 



\p+°-ie ->■ Jri 



12.8) 
;i2.9) 



12.1 A Gamma Decay 



Gamma-Ray Radiation 












Gamma Rays 



Parent Nucleus 
Cobalt -60 



Daughter Nucleus 
Ni-60 



• nucleus emits a 7 ray (a high energy photon) 

• often from an excited daughter nucleus from a a or (3 decay. 

• nuclei have energy levels 

• excited nuclei are marked by an asterix 

• mass numbers do not change 

tlNi*^llm + 1 (12.10) 



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12.1.5 Example 

the only stable isotope of cesium is Cs-133. Cs-137 is an unstable byproduct of nulcear 
power. Its decay often leads to an excited Barium nucleus, that emits a 7 ray. 

We notice that Cs-137 has too many neutrons (55p) so it decays by emission of a 



12.1.6 Radiation Penetration 

The pentration of radioactive particles is important for shielding and other uses, such 
as the use of 7 rays to sterilize food, and using absorption to control the thickness of 



• a-particles 

— Doubly charged, massive 

— very slow 

— stopped by a few centimeters of air or a sheet of paper 

• /3-particles 

— less massive 

— stopped by a few meters of air or milimeters of aluminium 

• 7 rays 

— can penetrate several centimeters of dense materials like lead Pb 

— Lead is commonly used as shielding for x-rays or gamma-rays 

— Removed/lose energy by compton scattering, photoelectric effect, pair pro- 
duction. 

• Damage from radiation is from the ionization of living cells 

• Continually exposed to normal background radiation 

• Workers in nuclear industries are monitored 

• only a small number of unstable nuclides occur naturally 

• U - 238 Pb - 206 after many steps 




Ba* -> 56 7 5a + 7 





metals. 



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12^2 Half-life 

Nuclei decay randomly at a rate characteristic of the particular nucleus. We can 
determine how many decays in a sample for given time, but not when a particular 
nucleus will decay. 

• activity (R) number of clear disintegrations per second. 

• Activity decreases with time 

AN 

=- = -\N (12.13) 
At v 1 

AN 

R=\ — \ = \N (12.14) 

N = N e- lambdat (12.15) 

where N is the number of atoms, N Q is the inital number, A is the decay constant. 

Decay rate is often expressed in half-life, or the time it takes for half of the sample 
to decay. 

N 1 . . 

ivT2 < m6 > 

After one half-life the activity is cut in half. 



12.2.1 Example: Sr-90 

the half-life of Sr — 90 is 28yrs. If we have lOOfig after 28yrs there is 50/ig of Sr 

TsSr^Y+^e (12.17) 

Half life of u-238 is 4.5 Billion years if the universe is 10 Billion years old about 
1/2 the original amount is still around, 
we can find the half life from, 

e- xt (12.18) 
-At (12.19) 

(12.20) 

Units: 

• ICi = 3.70 x 10 w Decays/s 

• IDecay/s = lBq (Bq=Bequeral) 



N _ 1 
N~o ~ 2 

0.693 

ti/2 = 



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2 4 6 8 10 

Time [s] 

12.2.2 Radioactive Dating 

• Because decay rates are constant they can be used as clocks 

• we can caculate backwards into time 

• C-14: Living objects have a constant amount of C-14 equal to the amount in 
the atmosphere, if we assume the amount of carbon in the atmosphere has not 
change appreciably then we can calculate the age of something by the amount 
of carbon 14. 

• current density 1 C - 14 per 7.2 x 10 n C - 12 

• ti/2 = 5730yrs 

• C-14 is produced by cosmic rays 

p+ -> n (12.21) 

l 4 N + n ->■ \ A C+\H (12.22) 

l 4 C -> } 4 N + p- (12.23) 

• intensity of cosmic rays are not constant 

• when organism dies 14 C starts to decay 

• Conentration of 14 C can be used to date the death. 



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Table 12.1: Nuclear Stability 
Number of stable isotopes Z N 
168 even even 

107 even(odd) odd(even) 

4 odd odd 



12.2.3 Example: Carbon 14 dating 

If there are 20 decays/min in lOg of carbon or 2.0decay/(gm). If the half life in a 
living organism is 16 decays/gm. How long ago was the organism living? 

16* V 1 / 2 8 V 1 / 2 4 V 1 / 2 2 (12.24) 
3(i 1/2 ) = 3(5730yr) = 1.7 x 10 4 y (12.25) 



12.3 Nuclear Stability 

• Depends on dommance of nuclear force over repulsive forces. 

• A < 40 N/Z « 1 

• A > 40 iV/Z > 1 

• Radioactive decay occurs until the nucleus lands on the stability curve 

Often times there is pairing between similar nucleons, look at the number of stable 
isotopes and the number of protons and neutrons, 
Criterial for stability; 

1. Z > 83 unstable 

2. Pairing 

(a) most e-e are stable 

(b) many o-e or e-o are stable 

(c) only 4 o-o are stable \H, %IA, fBe, \ A N 

3. number of nucleons 

(a) A < 40 Z « N 

(b) A > 40 N > Z 



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o a* 4Q *g> M 



Number of protons, Z 



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12.3.1 Binding Energy 

• can be calculated by the mass-energy equivalence 

• atomic mass unit (u) lu = 1.66 x lCT 27 /cg 

• E of one mass unit, 

E = mc 2 (12.26) 
= (1.66 x 10~ 27 %) (2.9977 x 10 8 m/s 2 ) = 1.4922 x 1(T 10 J (12.27) 
= 931.5MeV (12.28) 

Let us calculate the binding energy in terms of hte energy of hydrogen. Let us 
calculate the mass of He and compair it to the mass of 2 hydrogen. 

• mass of 2 H= 2.01565u 

• mass of 2 n = 2.07330u 

• sum of 2H + 2n=4.03980u 

• mass of 4 He =4.002603u 

• the mass difference Am = 4.03980m - 4.002603m = 0.030377m 

• Converting the mass difference to energy gives the binding energy, E = 28.30MeV^ 
Amc 2 



• Average per nucleon, 

E B 28.30Me\/ 



= 7.075MeV/nucleon (12.29) 



A 4 

• Much stronger than the binding energy of electrons. 

• If a massive nucleus is split (fissioned) the nucleons on the daughter nuclei are 
more tightly bound and energy is released 

• if 2 light nuclei are fused (fusion) the daugther nuclei is more tightly bound and 
energy is released 

• E B oc 1/A except for light nuclei 

• This implies nearest neighbor interactions 

There are magic numbers, if Z=2,8,20, 28, 50, 82, 126, if an element has a magic 
number of Z many isotopes are stable. 



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30 60 90 120 150 180 210 240 270 

Number of nucleoids in nucleus 

Figure 12.1: Nuclear Binding Energy 

12.4 Radiation Detection 

We can detect radiation by, 

• a- (3 transfer energy by electrical interactions. 

• 7 compton scattering, photo-electric effect 

• Particles produced by the interactions above can be detected. 

12.4.1 Gieger Counter 

• Hans Gieger (1882-1945) Student of Ernest Rutherford 

• Ionization of a gas creates a current event which is amplified and detected 

12.4.2 Scinillation Counter 

Atoms of phosphor material (Nal) is excited the excitation is measured by a photo- 
electric material (Photo-multiplier tube) 

12.4.3 Semi-conductor Detector 

Charged particles stike a semiconductor producing electrons which can be measured 



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Side View 



Gieger-Mudler Tube /" 
in Operation 

/ 

/ 

... 



Inner d:. 11 -r 
(.mode) 



Electron 
Cascades 

_ - Volts 



+ Volts 



Ionizing 
Radiation 

i 

I 

/ 



OuiscCcnducDO!: 



Lou- DensiLy Gii 



Figure 12.2: Geiger-Meuller tube 



Incident 
photon 



Photocathode 



Electrons 



Anode 




Electrical 
connectors 



X" 5SSS D/node 



Photomultiplier tube (PMT) 



Figure 12.3: Photomultiplier tube 



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Be Window 



\ 

] 



Ali Contact 
Dead Layer 



1/ 



incident 1 =hv , 
X-ray n 



escaped 
Si-KcT 
X-ray 



elecli-jn 



inner-shell 
ionization 




holes & 
photoelectrons 
(3.fl eV/pair) 



Figure 12.4: Si-Li Detector 

12.4.4 Cloud Chamber 

• Shows trajectory of the particle 

• cloud or bubble chamber created when a vapor or a liquid is super cooled by 
varying the volume and the pressure 

• spark super heated vapor created by varying the volume and the pressure 



12.5 Biological Effects and Medical Applications 

• Radiation Dosage: 

— Rotegen (R): number of x-rays or 7-rays required to produce an ionization 
charge of 2.58 x 10" 4 C/% 

— rad -1 rad - absorbed dose of lO~ 2 J/kg 

— Gray (Gy) - Si Unit 1 Gy=100rad 

• ionization of water on a cell can damage or kill a cell. 

• Effective dose : Depends on the type of radiation 



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© 



Figure 12.5: Cloud Chamber schematic 




Figure 12.6: Ions in a cloud chamber 



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— rem (rad equivalent man) 

— RBE relative biological effectiveness 

— RBE for x-ray, gamma-ray RBE=1 

— rem = rad x RBE 

— 20 rem a (lrad) same damage as 20 rem x-ray (20rad) 

— SI unit (Sv) 

12.6 Nuclear Reactions 

Nuclear reactions occur naturally when a nuclei decays into new nuclei. 

The first artificially produced nuclear reaction was by Rutherford, who bombared 
neutrons with alpha particles to produce a proton. 

One example of a nuclear reaction, 

\ 4 N +iHe^l 7 0+\H (12.30) 
The general form of a reaction is 

A + a^B + b (12.31) 

or A(a,b)B 

Atoms that are not stable can be created through nuclear reactions (Z > 83) 
12.6.1 Conservation of mass-energy 

In every nuclear reaction, the total relativistic energy must be conserved. 

(K N + m N c 2 ) + (K a + m a c 2 ) = (K + m c 2 ) + (K p + m N p 2 ) (12.32) 
K + K p — K N — K a = {m N + m a — m — m p )c 2 (12.33) 

The Q value is defined by the change in kinetic energy, 

Q = AK = (K + K p ) - (K N + K a ) (12.34) 
= (mjv + m a — mo — m p )c 2 (12.35) 

Thus for a reaction we can have mass changing into kinetic energy and vice versa. 

• if mass increases K I 

• if mass decreases K ~\ 



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If Q < the reaction requires a minimum amount of energy to occur. The Q for 
u N(a,p) 17 0. 

Q = {rriN + m a — mo — rn p )c 2 (12.36) 
= (14.003074m + 4.002603m - 16.99913m - 1.07825w)c 2 (12.37) 
= (-0.001281m)c 2 (12.38) 
= -1.193MeV (12.39) 

• Endoergic Q < 0, K — > m 

• Exoergic Q > 0, m — >• K 

Radioactive decay Q > (always) if Q < the minium kinetic energy required is, 

K mm = (1 + ^) | Q | (12.40) 

M A 

the minimum K is greater than Q because of the conservation of linear momentum. 
12.6.2 Example 

When a K m i n greater than the values for several possible reactions the result is gov- 
erned by the probability from Quantum Mechanics. The reaction's cross-section is a 
measure of the probability. 



12.7 Nuclear Fission 

In early attempts to make heavy nuclei, neutrons were shot at uranium atoms, some- 
times when a neutron strikes a U-238 molecule the nucleus will split forming nuclei 
of lighter elements. This is known as nuclear fission. 

In a fission reaction, a heavy nucleus splits into two smaller nuclei plus some 
energy. Spontaneous fission is a slow process. Fission can be induced. Induced fission 
is the basis of nuclear power and nuclear bombs. 



IfU+ln^QfU) lfXe+ltSr + 2C n) (12.41) 

lt 1 Ba+HKr + 3(ln) (12.42) 

ll Nd+HGe + 5C n) (12.43) 

• Only certain nuclei can undergo fission 

• Probability depends on the speed of neutrons 

- Slow neutrons K m leV 235 U, 239 Pu 



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- Fast neutrons K > lMeV 232 Th 

• E b /A curves give the estimate of the energy released 

- 235 U -> 

• Much more energy released than burning coal 

• Chain reaction : the neutrons released cause future fissions 

• the number of neutrons doubles w/ each generation, energy growth is expoential 

• To maintain a chain reaction you must have a minimum amount of fissible 
material 

- Crtical mass 

- At least one nuetron creates a new event 

• 235 f/, 23S U 

- 238 U can absorb neutrons and not fission 

- 235 U about 0.7% of naturally occuring uranium 

- Enriching process concentrates 235 U to reduce the critical mass 

* 3% to 5% reactor grade 

* 99% Weapons grade 

• If a chain reaction is not controlled, we have an explosion 

- A nuclear bomb maybe created by joining several sub-critical masses into 
a super-critical mass. 



12.7.1 Nuclear Power 

Fuel rods of enriched uranium are seperated by control rods (boron-cadmeium) and 
immersed in water. The reaction heats the water which intern is used to heat uncon- 
taminated water to steam which turns a turbine just like a coal plant. 

The control rods remove neutrons to slow the reaction. Fully inserted they stop 
the reaction. 

The core must always be cooled because of spontaneous decay. Water also acts 
as a moderator the speed of the emitted neutrons are slowed by the water, often this 
enables the chain reaction. 



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Cor crete shield 



Steel pressure s tea m generator 
vessel 

Control rods Pressuriser 



Steam 




Figure 12.7: Nuclear Reactor 



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12.7.2 Breeder Reactor 

238 U + fast neutrons 

In + 238 U U* -> 2 f Np + (3- -> 2 9 f Pu + 0- (12.44) 

94 9 Pu is a fissionable product. A breader reactor uses this reaction to generate pluto- 
nium for nuclear reactors (or bombs). Development in the US stopeed in the 1970's 
but it is big in france. 

12.7.3 Safety 

• LOCA : loss of coolant accident 

- coolant fails 

- rods heat-melt-fracture 

- fissing mass fall to floor into coolant causing steam or hydrogen explosion 

• meltdown -melt can fall to the floor and melt through the floor 

• TMI partial release of radioactive steam 

• Chernobyl Meltdown - LOCA caused by human error and bad design and a 
huge explosion 

• Where is waste stored 



12.8 Nuclear Fusion 

A fusion reaction is one where 2 light nuclei form a heavier nucleus. 

\H+\H ->■ \He +1 n + 17.QMeV (12.45) 
(2D){D-D) ->• 3.27MeV (12.46) 

A fusion reaction will release less energy per-reaction than a fusion reaction, but 
more energy per kg. The sun is powered by a fusion reaction. 

\H +\ H ->l H + /3 + v (12.47) 

The nuetrino is a subatomic particle generated by fusion reactions. 

\H+\H ->■ |ife + 7 (12.48) 

lHe+lHe ->■ iHe+\H+\H (12.49) 

If we summarize these reactions, 

A(\H) ^ H e + f3 + + 2 7 + 2v + Q (12.50) 

Q=+24.7MeV 



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Chapter 12 Introduction to Modern Physics: Physics 311 

12.8.1 Example: Fusion and the sun 

given: / = 1.40 x 10 3 W/m 2 striking the earth, what is the mass loss per second? 
given: R E _ S = 1.50 x 10 8 £;m = 1.5 x 10 n m 
M s = 2.00 x I0 30 kg 



A = 4%R 2 E _ S = 2.83 x 10 2 3m 2 

P s = IA = 3.96 x 10 26 J/s = 2.48 x 10 39 MeV/s 

Am (2.48 x 10 39 MeV/s) (1.66 x 10' 27 kg /u) 

~At ~ 931.5Mev/u 

12.8.2 Fusion Power 

• Need D 2 whiclh occurs naturally in the oceans. 

• less release of radioactive material than a fission reaction. 

• Shorter half-life 3 H t±/2 = 12.3yrs 

• Technically no method yet to control 

• Hydrogen bomb 

• Gas of positively charged ions = plasma 

• How to confine plasma in a specific density 

— Magnetic confinement 

* E-Field — > current — > heat (Tokomak) 

* minimum density and confinement time 

— Inertial Confinement 

* Pulses of laser explode H-pellets 

* Currently lasers are too week 

12.9 Neutrino 

The neutrinos introduced in order to conserve energy and linear momentum. 

• Usually some missing energy 

• Another particle is need, the neutrino 



(12.51) 
(12.52) 

(12.53) 



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0.2 0.4 0.6 0.8 1,0 1.2 



Kinetic energy, MeV 



Figure 12.8: Maximum Kinetic energy 



The neutrino interacts through the weak nuclear force 

l A C ->• \ A N + p- + V e (12.54) 
j 3 N ->■ l 3 C + /3 + + v e (12.55) 



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Chapter 13 



Quantum Mechanics VI: The 
Standard Model 



13.1 Beta Decay and neutrino 

13.2 Fundamental Forces 

The Heisenberg uncertainty principle AE oc -h allows for short term violation 
energy conservation, or the reaction happens so fast that it is not observed. This 
carried by a virtual or non-observed exchange particle. 
Know forces in order of decreasing strength 

1. Strong Nuclear Force 

2. Electro-magnetic Force 

3. Weak nuclear force 

4. gravitational force 
Particles that interact via strong 

• Hadrons - make of quarks 

— Baryons half integer spin -3 quarks 

— mesons integer spin - quark + anti-quark 

Particles that interact via week 

• Leptons 

— electron 

— muons 

— neutrinos 



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Chapter 13 Introduction to Modern Physics: Physics 311 

13.3 Electro-Magnetic force and the Photon 

The electro-magnetic forces interact via a virtual particle. 




Figure 13.1: Feynman diagram of photon exchange 



Change in momentum and energy of the particles is due to the virtual photon 
transfer. 

13.4 Strong Force and mesons 

In 1935 Hideki Yukawa (1907-1981) theorized a short range strong Nuclear force that 
held together the nucleus. The exchange particle that mediates the strong force is 
known as the meson. 



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The meson is a violation of the conservation of energy, by an amount of AE = 
m m c 2 , this means that meson has to be absorbed in a time, 

h h 

At w — — R < cAt = m m w 270m e (13.1) 

2nAE 1nm m c K ' 

Real mesons can be created by the collision of nucleon-nucleon interaction. Mesons 
were first observed in connection with cosmic rays in 1936. fi mu meson or a muon. 
In 1947, the pion n meson was discovered. 

7T ->■ fj+ + (13.2) 

v a muon neutrino. 



13.5 Weak nuclear force - W particle 

Enrico Fermi discovered the W particle, that is created during nuclear decay. Free 
neutrons typically decay in 10.4min. 

n ->■ p + + P~ + v e (13.3) 

The week nuclear force is very short range R ~ 10 _17 m, since the range is very small 
the carrier must have a large mass. 

The first observance of a non- virtual W occured in the 1980 's The weak force is, 

• weak is the only force on neutrinos 

• transmitting identities of particles in the nucleus. 

• Supernovas 



13.6 Gravity 

The mediating particle of the gravitational force is said to be the graviton. 

Experiment 15 — 200Ge^ electrons, muons, and neutrinos colliding with nucleons. 



Table 13.1: Fundamental Forces 



Force 




Range 


exchange 


Particles 


Fundamental Strong 


1 


w 10 _15 m 


7T 


quarks/gluons 


E-M 


10~ 3 


1/r 2 (oo) 


photon 


electrically charged 


Weak 


io- 8 


10- 17 m 


W, Z 


quarks/leptons 


Gravity 


10 45 


1/r 2 (oo) 


Graviton 


all 


Residual Strong 






Mesons 


Hadrons 



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gluon 



between quarks between nudeons 

Strong Interaction 

Figure 13.2: Feynman diagrams for the fundamental forces 

Table 13.2: Quarks 
up (+2/3e) top (+2/3e) strange (-l/3e) 
down (-l/3e) bottom (-l/3e) charm (+2/3e) 

13.7 Fundamental particles 

• Particles made of other particles, is there a limit? 

• Hadrons - particles that interact through the strong nuclear force, p, n, ir. 

• Leptons - interact through the weak force but not the strong force. 

e considered a point particle 

fi 

r tauon 

v neutrino very small mass. z/ e , z/^, v T 

• 1963 Gell-man, Zweig suggests that there are particles that make up hadrons. 
These particles are named quarks 

Fundamental particles of the Hadron Family. 
Current Elementary particles, Quarks and Leptons. 

Free quarks have not been observed. QCD, quarks given color to satisfy pauli 
exclusion. Force between colors gluons. 



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Chapter 13 Introduction to Modern Physics: Physics 311 

13.8 Unification Theories 

• Relativity unified gravitational, electro-magnetic and mechanics 

• 1960's Electro-weak force unified the electro-magnetic and weak forces - Nobel 
Prize 1979 Glashow, salazar, Weinberg. 

• GUT - Grand Unified Theory (not finished) to unify electro-weak and the strong 
force. 

The standard model is the electoweak force and quantum chromodynamics. 



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Part IV 

Applications of Quantum 
Mechanics 



127 



Chapter 14 
Quantum Statistics 



128 



Chapter 15 
Semiconductors 



129 



Chapter 15 Introduction to Modern Physics: Physics 311 



Appendices 



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Appendix A 

Nobel Prizes in Physics 



• 2008 - Yoichiro Nambu, Makoto Kobayashi, Toshihide Maskawa 

• 2007 - Albert Fert, Peter Grnberg 

• 2006 - John C. Mather, George F. Smoot 

• 2005 - Roy J. Glauber, John L. Hall, Theodor W. Hnsch 

• 2004 - David J. Gross, H. David Politzer, Frank Wilczek 

• 2003 - Alexei A. Abrikosov, Vitaly L. Ginzburg, Anthony J. Leggett 

• 2002 - Raymond Davis Jr., Masatoshi Koshiba, Riccardo Giacconi 

• 2001 - Eric A. Cornell, Wolfgang Ketterle, Carl E. Wieman 

• 2000 - Zhores I. Alferov, Herbert Kroemer, Jack S. Kilby 

• 1999 - Gerardus 't Hooft, Martinus J.G. Veltman 

• 1998 - Robert B. Laughlin, Horst L. Strmer, Daniel C. Tsui 

• 1997 - Steven Chu, Claude Cohen-Tannoudji, William D. Phillips 

• 1996 - David M. Lee, Douglas D. Osheroff, Robert C. Richardson 

• 1995 - Martin L. Perl, Frederick Reines 

• 1994 - Bertram N. Brockhouse, Clifford G. Shull 

• 1993 - Russell A. Hulse, Joseph H. Taylor Jr. 

• 1992 - Georges Charpak 

• 1991 - Pierre-Gilles de Gennes 



131 



Chapter A Introduction to Modern Physics: Physics 311 

• 1990 - Jerome I. Friedman, Henry W. Kendall, Richard E. Taylor 

• 1989 - Norman F. Ramsey, Hans G. Dehmelt, Wolfgang Paul 

• 1988 - Leon M. Lederman, Melvin Schwartz, Jack Steinberger 

• 1987 - J. Georg Bednorz, K. Alex Mller 

• 1986 - Ernst Ruska, Gerd Binnig, Heinrich Rohrer 

• 1985 - Klaus von Klitzing 

• 1984 - Carlo Rubbia, Simon van der Meer 

• 1983 - Subramanyan Chandrasekhar, William A. Fowler 

• 1982 - Kenneth G. Wilson 

• 1981 - Nicolaas Bloembergen, Arthur L. Schawlow, Kai M. Siegbahn 

• 1980 - James Cronin, Val Fitch 

• 1979 - Sheldon Glashow, Abdus Salam, Steven Weinberg 

• 1978 - Pyotr Kapitsa, Arno Penzias, Robert Woodrow Wilson 

• 1977 - Philip W. Anderson, Sir Nevill F. Mott, John H. van Vleck 

• 1976 - Burton Richter, Samuel C.C. Ting 

• 1975 - Aage N. Bohr, Ben R. Mottelson, James Rainwater 

• 1974 - Martin Ryle, Antony Hewish 

• 1973 - Leo Esaki, Ivar Giaever, Brian D. Josephson 

• 1972 - John Bardeen, Leon N. Cooper, Robert Schrieffer 

• 1971 - Dennis Gabor 

• 1970 - Hannes Alfvn, Louis Nel 

• 1969 - Murray Gell-Mann 

• 1968 - Luis Alvarez 

• 1967 - Hans Bethe 

• 1966 - Alfred Kastler 



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• 1965 - Sin-Itiro Tomonaga, Julian Schwinger, Richard P. Feynman 

• 1964 - Charles H. Townes, Nicolay G. Basov, Aleksandr M. Prokhorov 

• 1963 - Eugene Wigner, Maria Goeppert-Mayer, J. Hans D. Jensen 

• 1962 - Lev Landau 

• 1961 - Robert Hofstadter, Rudolf Mssbauer 

• 1960 - Donald A. Glaser 

• 1959 - Emilio Segr, Owen Chamberlain 

• 1958 - Pavel A. Cherenkov, Ilja M. Frank, Igor Y. Tamm 

• 1957 - Chen Ning Yang, Tsung-Dao Lee 

• 1956 - William B. Shockley, John Bardeen, Walter H. Brattain 

• 1955 - Willis E. Lamb, Polykarp Kusch 

• 1954 - Max Born, Walther Bothe 

• 1953 - Frits Zernike 

• 1952 - Felix Bloch, E. M. Purcell 

• 1951 - John Cockcroft, Ernest T.S. Walton 

• 1950 - Cecil Powell 

• 1949 - Hideki Yukawa 

• 1948 - Patrick M.S. Blackett 

• 1947 - Edward V. Appleton 

• 1946 - Percy W. Bridgman 

• 1945 - Wolfgang Pauli 

• 1944 - Isidor Isaac Rabi 

• 1943 - Otto Stern 

• 1942 - The prize money was with 1/3 allocated to the Main Fund and with 2/3 
to the Special Fund of this prize section 



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• 1941 - The prize money was with 1/3 allocated to the Main Fund and with 2/3 
to the Special Fund of this prize section 

• 1940 - The prize money was with 1/3 allocated to the Main Fund and with 2/3 
to the Special Fund of this prize section 

• 1939 - Ernest Lawrence 

• 1938 - Enrico Fermi 

• 1937 - Clinton Davisson, George Paget Thomson 

• 1936 - Victor F. Hess, Carl D. Anderson 

• 1935 - James Chadwick 

• 1934 - The prize money was with 1/3 allocated to the Main Fund and with 2/3 
to the Special Fund of this prize section 

• 1933 - Erwin Schrdinger, Paul A.M. Dirac 

• 1932 - Werner Heisenberg 

• 1931 - The prize money was allocated to the Special Fund of this prize section 

• 1930 - Sir Venkata Raman 

• 1929 - Louis de Broglie 

• 1928 - Owen Willans Richardson 

• 1927 - Arthur H. Compton, C.T.R. Wilson 

• 1926 - Jean Baptiste Perrin 

• 1925 - James Franck, Gustav Hertz 

• 1924 - Manne Siegbahn 

• 1923 - Robert A. Millikan 

• 1922 - Niels Bohr 

• 1921 - Albert Einstein 

• 1920 - Charles Edouard Guillaume 

• 1919 - Johannes Stark 

• 1918 - Max Planck 



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• 1917 - Charles Glover Barkla 

• 1916 - The prize money was allocated to the Special Fund of this prize section 

• 1915 - William Bragg, Lawrence Bragg 

• 1914 - Max von Laue 

• 1913 - Heike Kamerlingh Onnes 

• 1912 - Gustaf Daln 

• 1911 - Wilhelm Wien 

• 1910 - Johannes Diderik van der Waals 

• 1909 - Guglielmo Marconi, Ferdinand Braun 

• 1908 - Gabriel Lippmann 

• 1907 - Albert A. Michelson 

• 1906 - J.J. Thomson 

• 1905 - Philipp Lenard 

• 1904 - Lord Rayleigh 

• 1903 - Henri Becquerel, Pierre Curie, Marie Curie 

• 1902 - Hendrik A. Lorentz, Pieter Zeeman 

• 1901 - Wilhelm Conrad Rntgen 



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Appendix B 



Maxwell's Equations in the 
integral form 



fs En 



dA = —Q inside (B.l) 



B n dA = (B.2) 
Edl = f B n dA (B.3) 



dt j , 



j>Bdl = hqI + ^aj t J g E n dA ( B - 4 ) 



B.0.1 Maxwell's Equations in the derivative form 



V-E = -p (B.5) 



-O 



V-B = (B.6) 

Vx£ = =-- gr (B.7) 

VxB = fi J + fioe -^r ( B - 8 ) 
in regions where there is no charge or current, we write Maxwell's equations as, 

V-E = (B.9) 

V-fi = (B.10) 
f) R 

Vx£ = =-8f < BU > 

VxB = /i e — (B.12) 

136 



Chapter B Introduction to Modern Physics: Physics 311 

Now we apply the operator Vx to the second two equations to get, 

V x (V x E) = V(V • E) - V 2 E = Vx(— ) 

d ,„ d 2 E 
= -^(VxS) = -^ w 

since V • E — 0; 

n2r d 2 E 

V E = ^Oto-r^ 

and similarly, 

dE 

Vx(Vx5) = V(V-5)-V 2 5 = Vx(-) 

d d 2 B 

= -^x«) = -^ 

since V • -B = 0; 

<9 2 5 

Which is the same as the wave equation, 
the solution of which is, 

/ = f sin(kx - w£) 

Where the velocity v, 

1 

c = 



is same as the speed of light c = 3.00 x 10 8 m/s, and 

k - 2pi 
u = 2nf 



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Appendix C 



Alternate Derivations for Special 
Relativity 



C.l Building a suitable transformation: Einstein's 
way 

Imagine a coordinate systems K with points (x, y, z, t) and K' with points (£, 77, (, r) 
moving with velocity v relative to K. At a given time, t, the origins of two coordinate 
systems are coincident, the axes are aligned and the velocity of K' is in the x,£ 
direction. A light ray is emitted from the origin of k and reflected off of a point £' in 
k. 

For every point (x, y, z, t) there is a transformation to (£, 77, (, r). 

We make a note that x' is the distance from the origin O' to £' as seen in the 
frame K. (this is the same as the length of a moving rod measured in the stationary 
frame.) x' can be expressed as a function of time given by x' = x — vt 

Because the origin in K' (O r ) and $J are stationary with respect to each other the 
condition of simultaneity holds for K' \ 

l(T + T 2 ) = n (C.l) 

for a general transformation r can be a function of (x 1 , y, z, t), so we write the condi- 
tion of simultaneity as, 

\ [r(0, 0, 0, f ) + r(0, 0, 0, t + — + -^—)] = r(x', 0, 0, * + — ). (C.2) 
2 c — v c + v c — v 

If x' is very small we obtain (as differentiation), 

2 v c-v c + w ; at dx' c-vdt K 
Or v Or 

^ + ^m = < c ' 4 > 



138 



Chapter C 



Introduction to Modern Physics: Physics 311 



Homework 29 

Derive: Eq (C.4) from Eq (C.3). 
Solution: 

2^7^, + 7+^ ) + 7^, = 2I c 2 -v 2 ') (C - 5) 

(C.6) 
(C.7) 



1 


^-(c + u) 


- (c-v) + 2(c + v) 


2 




c 2 — V 2 


1 




■ c + v + 2c + 2v)\ 


2 


^— c — v - 


c 2 — V 2 J 


1 


2v 


V 


2 


c 2 — V 2 


c 2 — V 2 



Eq (C.4) implies a transformation of the form, 



r = a(t- -y^x') (C.8) 

Where a is a yet unknown function of the velocity. Now make the substitution 
x' — x — vt to get, 

r = a(t-^^) (C.9) 
c 2 — v 2 

cH-vh-vx + vH c 2 (t-^x) 
= < ) = <—T—^) (C- 10 ) 

" (t-~ 2 x) (C.11) 



1-4' C 2 



Similarly, in the x, ^-direction. 



£ = cr (C.12) 



£ = ac(t--2 (C.13) 

— V 2 

make the substitution that the time required to move the distance x' is, 



x' 



t= , (C.14) 

c — v 



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to get, 



£ = ac(t-^- 2 x'), (C.15) 
cr — v z 



x' 



= ac( - 2 2 x'), (C.16) 

c — v c l — V 1 

x'c + x'v-x'v 

= ac ( — — )' ( c - 18 ) 

= ac(^- 2 ), (C.19) 

C — V 



ac 2 



c 2 — V 2 



x', (C.20) 



Now substituting, x' = x — vt we obtain, 



ac 2 



£ = 3— (C21) 



c 2 — v 2 



Homework 30 

Derive Eq. (C.20) from Eq. (C.15) and the facts that t = and x' — x — vt. 

Similarly in the directions perpendicular to the relative movement of the two 
reference frames, (y and z). 

T] = CT (C.22) 

For a pulse of light moving entirely in the 77 direction. Its vertical speed however 
in the reference frame K is given by, v in the x direction and c in the direction of 
the light ray (now appearing to travel at an angle to the x-direction), is from the 
Pythagorean theorem, 

u = Vc 2 - v 2 (C.23) 
where the time it takes the light ray to travel a distance y is given by, 

* = -f^=f ( c - 24 ) 



cr — v 



Thus, with x' = in the K' frame, we write r\ as, 



v 



rj = C T = ac(t - -x) (C.25) 



c 2 — V 2 



= «*-=L=) (C26) 
Vr — v z 



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Similarly, ( may be written as, 



The equations, 



C = ac(-=L=) (C.27) 



r = (C.28) 



1 - V c 2 



^2 



£ = -^-j(x-vt), (C.29) 
c 2 _ ^ 

77 = ac( 7 === ), (C.30) 

C = «c(-=L=) (C.31) 



give the Lorentz transformations 
Using the transforma 

forms may be written as, 



Using the transformations, 6(v) = , ° „ , and /3 = , 1 „ . The Lorentz trans- 



r = 


<£(u)0(f - 


v s 
cr 


e = 


<f>(v)/3(x — 


vt), 


77 = 


<f>(v)y, 




c = 


4>(y)z. 





(C.32) 

(C.33) 
(C.34) 
(C.35) 



Homework 31 

Find the transformations 0(f), and f3 that allow, Eqs (C.28)-(C.31) to be written as 
Eqs (C.32)-(C.35). 



Homework 32 

If we have a wave given from a pulse of light starting at coincident origins in k and 
K, then the wave if emitted in all directions should be spherical in both coordinate 
systems. Show, 

transforms via the Lorentz transformations to, 

£ 2 + 77 2 + C 2 = cV (C.37) 



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C.l.l The nature of (j>(v) 

We need to examine <p(v) in more detail, given the transformation of, 

r = 4>{v)P{t-^x), (C.38) 
cr 

Let us introduce a 3rd set of coordinates K" moving with velocity — v with respect 
to K' . (This will make it stationary with respect to K). Now, transforming r to t' 
gives, 

H = <t>(-v)(3(-v)(r + - 2 0, (C.39) 

Where and (3 are functions of v, from the form of (3 we know that it is invariant 
under directional changes in v so (3(v) = j3(—v). Now, we express t' in terms of t, 



t' = 0( 


-^(-^)(r+^), 
c z 


(C.40) 


= 0( 




(C.41) 


= 0( 


-v)<P{v)(3 2 (t-^x+^{x-vt)), 


(C.42) 


= 0( 


\ it \n7.( V V 1,2 \ 

-vmv)p [t - ^ + -X- -t), 


(C.43) 


= 0( 




(C.44) 


= 0( 




(C.45) 


= 0( 




(C.46) 


= 0( 


-v)(f>(v)t, 


(C.47) 






(C.48) 



Because k' is stationary with respect to K their clocks should be synchronized, 
and t' = t. similarly, the transform of position gives, 

x ' = (f>(- v )<l)(v)x, (C.49) 

And we conclude that, 

<f)(-v)<j)(v) = 1 (C.50) 

Examining a rod moving in the x-direction, whos long axis is in the y-direction. In 
its stationary frame (k), the end points of the rod fall at, (£i = 0, rji — 0, Ci = 0) and 
(£2 — 0, 772 = I, C2 — 0). It follows from the transformation that its end points in the 
K frame should fall at, (xi = vt, y\ — 0, z\ — 0) and (x 2 = vt, y 2 = z 2 = 0). Since 



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the direction of the velocity (+ or -) depends solely upon the bias of the observer we 
would expect the length in the moving frame be the same regardless of the direction 
of the velocity, 

<j>{v) <j>{-v) 
which leads to the conclusion with, Eq (C.50) that, 

(j){v) = (f)(- v ) = 1. (C.52) 
This allows us to write the Lorentz transformations as, 

(C.53) 

(C.54) 
(C.55) 
(C.56) 



T 


= P(t- 


v , 
c* X ' 




= 0(x- 


vt), 


V 


= y, 




C 


= z. 





and, 



t 


= P(r + 


- 2 a 

c z 


X 


= m+ 


vt), 


y 


= V, 




z 


= c 





(C.57) 

(C.58) 
(C.59) 
(C.60) 



Note: most textbooks use the notation 7 = , 1 ^ 2 instead of what we are calling 

2 2 

while the factor \ is given the name j3 — \. 



C.2 The Lorentz transformation as a rotation in 
the complex plane 

This derivation is taken from Pathria. 

In order to derive the Lorentz transformations, we make some observations, 

1. Cartesian w/ aligned axes 

2. Relative motion is defined in the direction of one axis x, x' axes, the velocity of 
S' with respect to S is v. 



3. Coincident at t — 0, r = 0, 



x = y = z = t (c.61) 
£ = V = C = t (C.62) 



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at t = a light pulse is emitted from the origin and forms a spherical wave front. 

x 2 + y 2 + z 2 - (ct) 2 = 0, (C.63) 
e + V 2 + C 2 - {cr) 2 = (C.64) 

The law of inertia must be valid for both reference frames, the transformation between 
the two should be of a linear, homogeneous type. 

V = <P(v)y, (C.65) 

<p(v) can only be a function of velocity because S' and S remain parallel, now consider 
a reference frame S" moving with velocity with respect to S', then, 

y" = 4>{v)y' = <t>{-v)<t>{v)y (C.66) 

then (f>(—v)(j>(v)—l, since y" = y. 

Since the motion is orthogonal to the direction of movement. Motion to the left 
should be indistinguishable from motion to the right. 

<f>( v ) = (t>{- v ) (C.67) 

Therefore phi(v) = ±1 Since there is no inversion of axes, we conclude that <fi(v) = 1. 
This leads us to write, 

y' = y (C.68) 

z' = z (C.69) 

this implies, 

x 2 - (ct) 2 = e- (cr) 2 (C.70) 

define 

ict = x A (C.71) 

icr = f 4 , (C.72) 

now we can write Eq. (C.70) as, 

xl + xl = ei+£ (C.73) 
Now apply x\ and rr 4 as coordinates in the complex plane, 

r = x \ + x\ (C.74) 
where r is the distance from (0,0) to the point. Since £ 2 + £f is in the same form. 

r = e i +a (C75) 



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Chapter C Introduction to Modern Physics: Physics 311 

The new axes may be obtained by a rotation in the plane, 

£1 = xi cos + £4 sin <p (C.76) 
£4 = — X\ sin0 + x 4 cos0 (C.77) 

is the angle of rotation and is a function of v. An object at rest in S' must have a 
velocity v in S. 

§1 = 0, (C.78) 

P = -ivc (C.79) 
dx 4 

ADD SOME ALGEBRA 

dx l gcos0 + sin0 



dx 4 sin0 + cos. 



This leads to 



Therefore we write, 



c 



1 - ^ 

c 



2 



I V 2 

r o 



This means we can write, 



x — vt 



C.3 Velocity Transform 

We can write dx and dt in terms of the primed variables as, 



(C.80) 



71) 

tan0=- (C.81) 



sin0 = ] V, ° (C.82) 



cos0 = 1 (C.83) 



(C.84) 



r = (C.85) 



= p(dx' + vdt') (C.86) 
= P{dt'+^-dx') (C.87) 



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Now we write u x as, 



u x = ^ (C.88) 
f3(dx' + vdt>) 

f3(dt' + %dx>) { } 

Kdt (C.90) 



u' x + v 



(C.91) 



To transform a velocity perpendicular to the relative motion, (y-direction) dy = dy', 



dy dy' 



dt' 



We can sum up the transformations as, 

u' + v 



(C.93) 
(C.94) 



«, = j^r, (C.95) 



2 "a; 



»* = «T+W (a97) 



< = < a98 > 
= W^Y (a99) 

"» = W^r (ai00) 

(C.101) 



2 



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Chapter C Introduction to Modern Physics: Physics 311 

C.3.1 Example: transforming the velocity of an object mov- 
ing at the speed of light 



Now, let us apply this to an object moving at c in frame S'. 



u' x = c (C.102) 

7/' -I- 1) 

u x = ^4rv> ( c - 103 ) 



c + v 

l + 4c' 

C + V 



(C.104) 
(C.105) 



C.3.2 Transformation of momentum and energy from one 
frame to another 



We can see that the reletivistic momentum is written as function of its relative velocity 
in frame, S'. 



P<=^= (O.106) 



while in frame S, we write the momentum as, 



P=^^ (C.107) 



1 u ' 



i 



We know the rules for converting velocites, but we need to be able to convert, \j\ — - c 



to Jl-$. 



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We begin with the velocity transformations, 





u' x + v 


(C.108) 


u x = 






Uy = 




(C.109) 




u z = 


u' z 


(C.110) 




u 'x = 


u x — V 


(C.111) 


l->*' 


u' y = 


My 


(C.112) 


0(l->*)' 






(C.113) 
(C.114) 


/0(l->*)' 



Then we can write u' 2 as, 



u 



12 



Ux + + z - (i - f,u x y + p(i - f 2Ux y + J 



u' 2 = 




1 


P 2 (l 




u' 2 




1 


c 2 


/? 2 (1 




u' 2 




1 


c 2 




~^2 U x) 2 



(f3 2 (u x -v) 2 + u 2 y + u 2 z ), 



(/3 2 (i->,) 2 -^K-^) 



,2 _ "7 

c 2 c 2 



(C.116) 
(C.117) 
), (C.118) 



Let us look at the u x term for a moment, 



(3 2 (l-^u x ) 2 -^(u x -v) 2 



= /? 2 

= /? 2 

= /? 2 ( 

= 1 



4 + *5E-%119) 
c 2 <r cr J 



c 2 c 2 



V 2 

+ — 



(C.120) 
(C.121) 
(C.122) 



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Substituing this result back into Eq (C.118), 

1-— - 1 ( 1 < u l u l\ (nioa) 

1 C 2 " /32 (1 _ « Ux )2 ^ C 2 C 2 C 2> 

/3(l-^g_L= (C.125) 



1 

c- 



' 2 C 2 ' /-, «2 



2 A/ x r 2 



now we define, 



</(«) = . (C.126) 



i y?_ 



We now write Eq (C.125) as, 

g{u>) = p(g{u)-g{u) V -^-) (C.127) 
g{u) = p(g(u')+g(u') V -^-) (C.128) 

and Eqs. (C.106) and (C.107) as, 

p' = m u'g(u'), (C.129) 
p = moug(u), (C.130) 

Now we have the math in place to complete the transformation of the momentum 
from the S' to the S frame. We will transform each carteasian coordinate seperately, 

p x = m u x g(u), (C.131) 
p' x = m u x g(u) (C.132) 

= m (-^f)/3,( M )(l-^), (C.133) 



v 



= m (u x - v)/3g(u) = /3(m u x g(u) m ug(u)), (C.134) 

u 

p' x = 0(p x --p). (C.135) 



Recall, 



(C.136) 



u pc 
c = E 

u = ^ (C.137) 



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Now we finally write, 

p' x = 0(p x --p), (C.138) 
u 

= p( Px -Y^E), (C.139) 

Transforming momentums in the y— direction (and z-direction) 

p y = m u y g(u), (C.140) 

p'y = rn u' y g(u') (C.141) 

Py = rn u y g{u) j^ _ (C.142) 

p' y = m u y g(u)=p y (C.143) 

Transforming the energy requires a bit more insight, E = mc 2 , start with the 
transform, 

g(u>) = p(g( u ) - g{u) V -^f) (C.144) 
multiply by m to find the transform between relativistic masses, 



g(u>) = p(g(u)-g(u) V -^), (C.145) 

/ V'Ur \ 

m g(u') = p\m g(u) -m g(u)—J, (C.146) 

m = p( m -^p x ), (C.147) 



9 ' 



E' „ (E 



= (c - 148) 



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